Receiver method, receiver, transmission method, transmitter, transmitter-receiver system, and communication apparatus

ABSTRACT

In accordance with an embodiment of the present invention, a method for receiving a signal, comprising the estimation step for estimating time and frequency shifts that are embedded in the received signal, to cancel-out shifts, wherein the method refers to the non-commutative shift parameter space of co-dimension 2.

CROSS REFERENCE TO RELATED APPLICATIONS

This Nonprovisional application is a continuation of PCT InternationalApplication No. PCT/JP2018/024592 filed in Japan on Jun. 28, 2018, whichclaims priority under 35 U.S.C. § 119 on PCT International ApplicationNo. PCT/JP2018/001735 filed in Japan on Jan. 22, 2018, the entirecontents of which are hereby incorporated by reference.

TECHNICAL FIELD

The present patent proposes for designing receive methods, receiveapparatus, transmitting method, transmitting apparatus, andtransmitter-receiver systems.

BACKGROUND FIELD

There are many kinds of techniques about communication systems; Thesetechniques have also led to technical changes in communication; Theinventor has been proposed a transmitter-receiver system usingtime-division and frequency-division schemes (e.g., Patent list [1]-[6]and Non-Patent list [1]-[32]):

CITATION LIST Patent Literature REFERENCES

-   [1] JT2016-189500-   [2] WO2012/153732 A1-   [3] WO2014/034664A1-   [4] WO2013/183722 A1-   [5] JP2016-189501A-   [6] JP2016-189502A-   [7] JP2013-251902A-   [8] JP2012-170083A

Non-Patent Literature References

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Daubechies, The wavelet transform, time-frequency    localization and signal analysis,” IEEE Trans. Information Theory,    36-5, 961-1005, 1990.-   [9] B. Le. Floch, M. Alard & C. Berrou, “coded OFDM: Coded    Orthogonal Frequency Division Multiplex,” Proc. of IEEE, 83-6,    982-996, 1995.-   [10] P. Siohan, C. Sciclet, & N. Lacaille, “OFDM/OQAM: Analysis and    Design of OFDM/OQAM Systems, based on Filterbank Theory,” IEEE,    Trans. Sig. 50-5, 1170-1183, May, 2002.-   [11] Sciclet, C., Siohan, P. & Pinchon, D. Perfect reconstruction    conditions and design of oversampled DFT-based transmultiplexers,    EURASIP J. on Applied Signal Processing, 2006, Article ID 15756,    1-14, 2006.-   [12] B. F. Boroujeny, “OFDM Versus Filter Bank Multicarrier,” IEEE    Signal Processing Magazine, 28-3, 92-112, 2011.-   [13] P. P. Vaidyanathan, “Multirate Systems and Filter Banks,”    Prentice-Hall, 1993.-   [14] J. Ville, “Theorie et application de la notion de signal    analytique,” Cables et transmission, no. 2, pp. 61-74. 1948. (J.    Ville: “Theory and Applications of the notion of complex signal”,    translated from the French by I. Stein, T-92, 8-1-58, U.S. Air Force    Project Rand, 1958)-   [15] P. M. Woodward, Probability and Information Theory, with    Applications to Radar, Pergamon Press, New York, 1953.-   [16] C. H. Wilcox, “The synthesis problem for radar ambiguity    function,” MRC Technical Report, No. 157, pp. 1-46. Mathematics    Reaearch Center, U.S. Army, Univ. Wisconsin, Madison, 1960.-   [17] L. Auslander and R. Tolimieri, “Radar Ambiguity Functions and    Group Theory, SIAM J. Math. Anal., 16-3, 577-601, 1985.-   [18] C. W. Helstrom, Elements of Signal Detection and Estimation,”    PTR Prentice-Hall, 1995.-   [19] N. Levanson and E. Mozeson, “Radar Signals,” Wiley    interscience, 2004-   [20] Sakurai, J. J. Modern quantum mechanics, S. F. Tuan editor,    Rev. ed., Addison-Wesley Pub. Comp. 1994.-   [21] J. von Neumann, The Geometry of Operators, vol. II (Ann. Math.    Studies, no. 22), 1950.-   [22] Youla, D. C Generalized image restoration by the method of    alternating orthogonal projections, IEEE Trans. Circuits and    Systems, CAS-25-9, 694-702, 1978.-   [23] Stark, H., Cahana, D.& Webb, H. Restoration of arbitrary    finiteenergy optical objects from limited spatial and spectral    information, J. Opt. Soc. Amer., 71-6, 635-642, 1981.-   [24] Kohda, T., Jitsumatsu, Y.& Aihara, K. Separability of    time-frequency synchronization, Proc. Int. Radar Symp., 964-969,    2013.-   [25] T. Kohda, Y Jitsumatsu, and K. Aihara, “Gabor division/spread    spectrum system is separable in time and frequency synchronization,”    Proc. VTC 2013 Fall, 1-5, 2013.-   [26] Y. Jitsumatsu, T. Kohda, and K. Aihara, “Spread Spectrum-based    Cooperative and individual time-frequency synchronization,” Proc.    (ISWCS), 1-5 2013.-   [27] Jitsumatsu, Y., Kohada, T. &Aihara, K. Delay-Doppler space    division-based multiple-access solves multiple-target detection,    Jonnsson, M., et al, (eds.) MACOM2013, LNCS8310, Springer, 39-53,    2013-   [28] T. Kohada, Y. Jitsumatsu, and K. Aihara, “Recovering    noncoherent MPSK signal with unknown delay and Doppler using its    ambiguity function,” 4th International workshop on recent Advanced    in Broadband Access NetWork, (RABAN2013), 251-256, 2013.-   [29] T. Kohda, Y. Jitsumatsu and K. Aihara “Phase-tuned layers with    multiple 2D SS codes realize 16PSK communication,” 2014 2014 IEEE    Wireless Commun. Networking Conference, WCNC 2014, 469-474 (2014).-   [30] Jitsumatsu, Y.& Kohda, T. Digital phase updating loop and    delay-Doppler space division multiplexing for higher order MPSK,    Jonnsson, M., et al, (eds.) MACOM2014, LNCS8715, Springer, 1-15,    2014.-   [31] T. Kohda, Y. Jitsumatsu, and K. Aihara, “Frequency-division    spread spectrum makes frequency synchronisation easy,” Proc. IEEE    Globecom 2012, 3952-3958, 2012.-   [32] T. Kohda, Y. Jitsumatsu, and K. Aihara, “Frequency    synchronisation using SS technique,” Proc. The ninth Int. Sympo. on    Wireless Communication Systems, 855-859, 2012.-   [33] J. F. Daughman, “Two-dimensional analysis of cortical receptive    field profiles,” Vision Research, 20, 846-856, 1980.-   [34] J. F. Daughman, “Image analysis by local 2-D spectral    signatures,” J. Opt. Soc. Amer. (A), 2, p. P74, 1985.-   [35] J. F. Daughman, “Complete Discrete 2-D Gabor Transform by    Neural Networks for Image Analysis and Compression,” IEEE Trans.    Acoustics, Speech and Signal Processing, 36-7, 1169-1179, 1988.-   [36] Movella, Javier R. “Tutorial on Gabor filters”. Archived from    on 2009 Apr. 19, Retrieved 2008 May 14.-   [37] Hans G. Feichtinger and Thomas Strohmer: Gabor Analysis and    Algorithms, Birkhauser, 1998.-   [38] Jones, J. P. and Palmer, L. A. “An evaluation of the    two-dimensional gabor filter model of simple receptive fields in cat    striate cortex”. J. Neurophysiol. 58(6): 1233-1258. 1987-   [39] Tai Sing Lee, “Image representation using 2d Gabor wavelets,”    IEEE Trans. on pattern analysis and machine intelligence, 18-10,    1-13, 1996.-   [40] W. D. Montgomery, “Optical applications of von Neumann's    alternating projection theorem,” Optics Letters, 7-1, 1-3, 1982.-   [41] W. D. Montgomery “Restoration of images processing a finite    Fourier series,” Optics Letters, 7-2, 54-56, 1982.-   [42] A. V. Oppenheim and J. S. Lim, “Importance of phase in    Signals,” 1980, Proc of the IEEE, 96-5, 529-541, 1981

SUMMARY OF THE INVENTION Problems to be Solved by the Invention

The inventor remarked that the time-frequency shift operator (TFSO)representing the non-commutative property (NCP) of time-frequency shift(TFS)s in the time-frequency plane (TFP) plays an important role insignal detection and estimation and concluded that the NCP of the TFSOshould be noticed.

To solve the above problem, in accordance with an embodiment of thepresent invention, a proposed method mainly depends on this remark andits aim is to give efficient receive method, receiver, transmissionmethod, transmitter, and transmitter-receiver system.

Solution to Problem

To solve the above problem, a method, in accordance with an embodimentof the present invention, for receiving a signal is a method forreceiving a signal, comprising an estimation step for estimating a timeshift and a frequency shift that are embedded in a received signal,wherein the estimation step refers to a non-commutative time-frequencyshift parameter space of co-dimension 2.

To solve the above problem, a receiver apparatus in accordance with anembodiment of the present invention is a receiver apparatus forreceiving a signal, comprising an estimation part for estimating a timeshift and a frequency shift that are embedded in a received signal, withreference to a non-commutative shift parameter space of co-dimension 2.

To solve the above problem, a method, in accordance with an embodimentof the present invention, for transmitting a signal is a method fortransmitting a signal, comprising a shift step for time-frequencyshifting the signal to be transmitted, with reference to anoncommutative time-frequency shift parameter space of co-dimension 2.

To solve the above problem, a transmitter apparatus in accordance withan embodiment of the present invention is a transmitter apparatus fortransmitting a signal, comprising a time-frequency shift part fortime-frequency shifting the signal to be transmitted, with reference toa non-commutative time-frequency shift parameter space of co-dimension2.

To solve the above problem, a method, in accordance with an embodimentof the present invention, for receiving an image signal is a method forreceiving an image signal, comprising an estimation step for estimatinga space shift and a spatial frequency shift that are embedded in thereceived image, with reference to a parameter space, wherein each of thespace shift and the spatial frequency shift has dimension 2.

To solve the above problem, a method, in accordance with an embodimentof the present invention, for transmitting an image signal is a methodfor transmitting an image signal, comprising a shift step forspace-spatial frequency shifting the image signal to be transmitted,with reference to a parameter space, wherein each of the space shift andthe spatial frequency shift has dimension 2.

Advantageous Effects of Invention

The present disclosure enables one to realize an efficient receivemethod, a receiver, a transmission method, a transmitter, and atransmitter-receiver system.

FIGURE DESCRIPTION

FIG. 1 In accordance with an embodiment of the present invention, this

Fig. shows three kinds of divisions of the time-frequency plane (TFP):(a) indicates time division, (b) frequency division, and (c) Gabordivision [1]. The solid lines in (a) show divisions of data of timeduration T; the thin lines subdivisions by Time Domain (TD)-Phase code(PC); The dotted lines in (b) show divisions of data of bandwidth F; thedashed lines subdivisions by Frequency Domain (FD)-PC.

FIG. 2 In accordance with an embodiment of the present invention, thisFig. shows an illustration of the non-commutative property (NCP) of timeand frequency shift (TFS)s: The NCP is manifested in the expression ofthe product of the shift operators

T_(τ,0)·T_(0,υ)=e^(−i2πτυ)T_(0,υ)T_(τ,0): Its LHS corresponds to thetriangle in the figure; the RHS to the square; the Phase Distortion (PD)e^(−i2πτυ) appears; the circle in the figure shows the SymmetricalTime-Frequency Shift Operator (TFSO)[26]

[Mathematical Formula (MF)1]

_(τ,υ)

e ^(iπτυ)

_(r,0)·

_(0,υ).

FIG. 3 In accordance with an embodiment of the present invention, a)indicates the Gabor function located on the TFP and its related issues:a0) shows Gaussian chip waveform g_(mm′)(t) located on the TFP and itsFourier Transform (FT) G_(mm′)(f); a1) indicates the real and imaginaryparts of the TD-template, the combination of g_(mm′)(t) weighted by theFD PC X′_(m′): a2) indicates the real and imaginary parts of theFD-template, the combination of G_(mm′)(f) weighted by the TD PC X_(m);b) shows NN′ Cross-Correlation Function (CCF)s, N′ column-sums of valuesof the TD-CCFs, and N row-sums of values of the FD-CCFs; c) shows thealternative projection, orthogonal projecting onto the Time-Limited TimeDomain (TL-TD) and onto Band-Limited Frequency Domain (BL-FD), based onthe Alternative Projection Theorem (APT), the estimated

[MF2]

{circumflex over (t)} _(d) ,{circumflex over (f)} _(D),

and convergent values t_(d), f_(D).

FIG. 4 In accordance with an embodiment of the present invention, thisFig. shows the Synthesis Filter Bank (SFB) that contains TD-, FD-PCsX_(m′)X′_(m′) and the m′-th TD-template

[MF3]

u _(m′) ^(TD)[k], 0≤m′≤N′−1,

and generates the TD signature v[k] in (25), (67).

FIG. 5 In accordance with an embodiment of the present invention, thisFig. shows the SFB that contains TD-, FD-PCs X_(m′)X′_(m′) and the m′-thFD-template

[MF4]

U _(m) ^(FD)[

], 0≤m≤N−1.

and generates the FD signature in (25), (67)

[MF5]

V[

].

FIG. 6 In accordance with an embodiment of the present invention, thisFig. shows the SFB with input, the complex-valued data with address p,p′

[MF6]

{d _(p,p′)}_(p,p′=1) ^(P,P′)

generating the associated output, TD-Complex Envelope (CE) in (27), (71)

[MF7]

ψ[k].

FIG. 7 In accordance with an embodiment of the present invention, thisFig. shows the SFB with input, the complex-valued data

[MF8]

{d _(p,p′)}_(p,p′=1) ^(P,P′)

generating the output, FD-CE in (27), (71)

[MF9]

Ψ[

].

FIG. 8 In accordance with an embodiment of the present invention, thisFig. shows the Analysis Filter Bank (AFB) that is an array of N′TD-cross-correlation (CCR)s for decoding the complex-valued data

[MF10]

{d _(p,p′)}_(p=1) ^(P), 1≤p′≤P′.

FIG. 9 In accordance with an embodiment of the present invention, thisFig. shows the AFB that is an array of N FD-CCRs for decoding thecomplex-valued data

[MF11]

{d _(p,p′)}_(p′=1) ^(P′), 1≤p≤P.

FIG. 10 In accordance with an embodiment of the present invention, a)indicates an array of N′ TD-CCRs, c) an array of N FD-CCRs, and b) anillustration that the maximum likelihood estimate (MLE)s are obtained bythese two types of arrays of CCRs and are alternatively updated oneanother by the von Neumann'a Alternative Projection Theorem (APT).

FIG. 11 In accordance with an embodiment of the present invention, thisFig. shows an illustration of the von Neumann's APT, where Time-LimitedTime Domain (TL-TD) and Band-Limited Frequency Domain (BL-FD) indicatetwo subspaces of the Hilbert space; The subspace TL-TD is an L Δt (orT_(s)) TL-TD; The subspace BL-FD is an L Δf (or F_(s)) BL-FD; the arrowin the figure means the orthogonal projecting onto the associatedsubspace; This results in getting the MLE and the cardinal numbers ofthe CCFs.

FIG. 12 In accordance with an embodiment of the present invention, thisFig. shows a block-diagram of the system consisted of two alternativetransmitters, controlled by switches, that perform efficient and jointestimation of delay and Doppler with high-precision, with (or without)being equipped with

[MF12]

-ary PSK communication and its transmitter (encoder) capable of jointlyestimating delay and Doppler efficiently and precisely.

FIG. 13 In accordance with an embodiment of the present invention, thisFig. shows a block-diagram of the system consisted of two alternativereceivers, controlled by switches, that perform efficient and jointestimation of delay and Doppler with high-precision, with (or without)being equipped with

[MF13]

-ary PSK communication and its receiver-synchronizer (decoder) capableof jointly estimating delay and Doppler efficiently and precisely.

FIG. 14 In accordance with an embodiment of the present invention, thisFig. shows an example of the distribution of values of the real part ofthe CCF as a function of the delay τ and Doppler υ of the Main Channel(MC).

FIG. 15 In accordance with an embodiment of the present invention, thisFig. shows an example of the distribution of values of the real part ofthe CCF on the delay τ-Doppler υ space when the Artificial Channel (AC)is added to the main channel (MC).

FIG. 16 In accordance with an embodiment of the present invention, thisFig. shows the division of symbol's time-frequency plane (TFP) using anon-commutative AC-shift parameter space of co-dimension 2. The divisionof the TFP S of time duration T_(s) and bandwidth F_(s) (i.e., the Gabordivision): (S⁽⁰⁾,S⁽¹⁾,S⁽²⁾,S⁽³⁾) shows a vertical axis perpendicularlyattached to it, with a scale of non-commutative AC shifts

[MF14]

(k _(d) ⁽⁰⁾,

_(D) ⁽⁰⁾),(k _(d) ⁽¹⁾,

_(D) ⁽¹⁾),(k _(d) ⁽²⁾,

_(D) ⁽²⁾),(k _(d) ⁽³⁾,

_(D) ⁽³⁾)

and their associated 2-D PC codes according to the division

[MF15]

χ^((i)).

FIG. 17 In accordance with an embodiment of the present invention, thisFig. shows a division of the TFP using a non-commutative shift parameterspace, where each of AC0-TFP, AC1-TFP, AC2-TFP, and AC3-TFP is shiftedby its associated non-commutative shift.

FIG. 18 In accordance with an embodiment of the present invention, thisFig. shows the block-diagram of transmitter and receiver apparatuses.

FIG. 19 In accordance with an embodiment of the present invention, thisFig. shows the flow-chart of signal processing in the transmitter andreceiver apparatuses.

FIG. 20 In accordance with an embodiment of the present invention, thisFig. shows the block-diagram of transmitter and receiver apparatuses.

FIG. 21 In accordance with an embodiment of the present invention, thisFig. shows the flow-chart of signal processing in the transmitter andreceiver apparatuses.

DESCRIPTION OF THE PREFERRED EMBODIMENT Embodiment 1

In one embodiment of the invention, referring to Figures, the inventorexplains the transmitter-receiver system. The inventor starts byexpressing the theoretical issues behind the proposed method and anexample of embodiment. Next, the inventor gives a correspondence betweenthe proposed method and the content of the scope of claims describedbelow.

This disclosure cites several references by referring to patentreferences and non-patent references and hence the citation is withinthis disclosure.

These references are listed to cite technical terms and to refer theproblems to be solved and the background, relating to this disclosure.Thus the citation does not affect the patentability of the presentdisclosure.

<<Summary of Theoretical View Points: Communication Exploiting the NCPof TFSOs—Designs of a Transmitter Capable of Jointly Estimating Delayand Doppler and Those of its Receiver->>

It is not easy to perform synchronization of TD- and FD-Divisioncommunication systems, that are designed to convey data symbol ofduration T_(s) and bandwidth F_(s) efficiently. The radar problem ofestimating delay t_(d) and Doppler f_(D) from a received echo signalremains unsolved. These difficulties come from phase distortion (PD)s

[MF16]

e ^(j2piT) ^(s) ^(F) ^(s)

and

[MF17]

e ^(j2pit) ^(d) ^(f) ^(D)

that are generated by TFSOs and are similar as well-known algebraicrelations for the position and momentum operators in quantummechanics[4].The basic purpose of a radar is to detect the presence of an object ofinterest, called target detection, and provide information concerningthe object's location, motion, and other parameters, referred to asparameter estimation. So, radars are based on the statistical testing ofhypotheses for signal detection and estimation. The determination ofdelay and Doppler is an estimation problem from the PDs containing twounknowns. Neither delay nor Doppler is successfully detected andestimated with high precision from a noisy received signal without thehelp of the Weyl-Heisenberg group (WHG) theory. The description of theinvention is summarized as the following 5 items.

(summary1) When one designs a transmit signal, one should treat a signalin the TD and its Fourier transform (FT), i.e., a signal in the FDsymmetrically. In addition, the symmetrical time and frequency shiftoperator (TFSO) satisfying time-frequency symmetrical property (TFSP) isproven to be an operator such that the address of a multiplexed signalis manifested by the non-commutative property (NCP) of TFSs (cf. (39),(44). (51), (56)).

(summary2) Time-frequency shifted Gaussian pulses, i.e., Gabor functionsby 2-D binary phase-shift-keying (BPSK) modulation by TD- and FD-PCs ofperiods N,N′ are shown to be useful for maximum likelihood estimate(MLE)s of parameters t_(d), f_(D) among N′,N hypotheses in FD- andTD-likelihood functional (LF)s, respectively.

TD- and FD-PCs are usually called “2-D spread spectrum (SS) codes (theadjective “SS” is something of a misnomer) to the contrary, the inventorcalls it a 2-D PC for BPSK modulation; Note that BPSK modulation has twofunctions: merits and demerits. Many researchers have not been reallyaware of the important roles of BPSK modulation. The present disclosuregives the pros and cons of the BPSK modulation that have not beennoticed as follows.

The TD- and FD-CEs (27), wideband signals, called “signatures”, containPDs due to the BPSK modulation. The modulation makes a situation that NTD-template CE of type-3 (29) (or of type-1 (49)) with its support[0,T_(s)]×[0,LΔf] (or [0,LΔt]×[0, F_(s)]) and N′ FD-template CE oftype-4 (33) (or of type-2 (54)) with support [0,LΔt]×[0, F_(s)] (or [0,T_(s)]×[0,LΔf]) are automatically embedded into these TD- and FD-CEs asindications of matching. Hence these PDs play an important role inhypotheses-testing by the use of CCRs (see (30), Proposition 4 and (35),Proposition 5).

(summary3) Phase information of a signal has not been effectively usedin ordinary MLEs to the contrary, the inventor defines 4 kinds of TD-and FD-cross-correlation function (CCF)s between a received signal andtemplates: TD-template CE of type 3 (or of type 1) and FD-template CE oftype 4 (or of type 2) as a kind of optimum receivers. These CCFs areproven to have rigorous expressions in the product form of the ambiguityfunction (AF) and several twiddle factors, defined by a discretizedsignal of time- and frequency-samplings Δt, Δf=(LΔt)⁻¹, whenco-operatively using BPSK modulation with TD- and FD-PCs, and denoted as

[MF18]

$W = {e^{\frac{{- j}\; 2\pi}{L}}.}$

The PDs due to the NCP of TFSs can be evaluated in the powers of thetwiddle factors, i.e., the summation of the PDs over the chip-addresscan be rigorously represented in the form of DFT and IDFT; Thus thisexpression has a product form of three functions (see (41) in Lemma 2,(45) in Lemma 4, and (52), (57)). In accordance with an embodiment ofthe present invention, an efficient computation by DSP can beguaranteed.

(summary4) Using the Youla's signal reconstruction method[22], theinventor gives a proof of the APT-based Phase Updating Loop (PUL)algorithm, defined and introduced in the patent reference, patent[1].

First, define T_(s) (or LΔt)-time limited (TL) TD space, E₃ (or E₁) andF_(s) (or LΔf)-band limited (BL) FD space E₄ (or E₂), as subspaces ofthe Hilbert space.

Secondly, according to the arrays of N′ TD-CCFs and N FD-CCFs, define 4projection operator (PO)s orthogonal projecting onto E₃ (or E₁) and ontoE₄ (or E₂), denoted by P₃ (or P₁) and P₄ (or P₂), respectively.

Thirdly, in accordance with an embodiment of the present invention, theinventor defines the alternative projection theorem operator (APTO)based on the alternative projection theorem (APT), defined as

[MF19]

₃

^(−1,d)

₄

^(d) (or

₄

^(d)

₃

^(−1,d)),

where F^(−1,d),F^(d) denote the IDFT, DFT. Fourthly, the inventor givesan expression (59) for updating MLEs of a gain factor Ae^(iκ) of thechannel as a function of estimates

[MF20]

{circumflex over (t)} _(d) ,{circumflex over (f)} _(D)

and two expressions (60), (61) for updating MLE of

[MF21]

{circumflex over (t)} _(d) ,{circumflex over (f)} _(D).

Using these three expressions for updating MLEs, the inventor concludesthat (t_(d), f_(D)) are estimated within the convergence region of theAPT operator, i.e., the rectangle of chip-pulse duration LΔt and chipbandwidth LΔf with chip and data addresses ((ρ,ρ′),p^(→)) and provesthat MLEs

[MF22]

{circumflex over (t)} _(d) ,{circumflex over (f)} _(D)

are estimated within LΔt×LΔf and its computional complexity order is

[MF23]

(N+N′),

in place of

[MF24]

(N·N′).

That is, this APT operator singles out some rectangle in thetime-frequency plane (TFP) and filters out other regions. Such anoperator is referred to as a phase-space (or time-frequency)localization operator[7] and thus plays an important role of filters, inplace of conventional sharp filters usually used in DSP.

A Gaussian function is not employed in most of communication systemsprimarily because it does not satisfy the Nyquist condition. However,several favourable properties of Gaussians in the TFP are shown to be ofbenefit to our (t_(d), f_(D))-estimation problem and hence Gaussians areshown to be of crucial importance.

The PUL algorithm is an iteration for searching (t_(d), f_(D)) with norestriction of the range of (t_(d), f_(D)) if the resource of timeduration PT_(s) and bandwidth P′F_(s), for the data-level address

[MF25]

{right arrow over (p)}=(p,p′)

can be available. Accordingly, it is shown that the use of a combinationof a transmitter of a 2-D PC modulated signal of TD-, FD-Gaussianfunctions and a receiver in which the PUL is implemented in TD-, FD-CCFarrays makes it possible to provide a communication system which iscapable of high-precision and high-speed parameter estimation. In otherwords, the use of the above-described configuration presents a paradigmshift in communication systems utilizing NCP.

(summary5) the inventor gives an encoding-decoding system for a high

[MF26]

-PSK communication in cooperation with establishing synchronization.

[MF27]

-PSK communication is available to an automotive radar capable of bothestimating delay-Doppler and communicating to another object with data.It is not easy to transmit an

[MF28]

-PSK signal

[MF29]

${\exp \mspace{11mu} \left( \frac{i\; 2\; \pi \; k}{\mathcal{M}} \right)},\left( {0 \leq k \leq {\mathcal{M} - 1}} \right)$

primarily because the identification of the phase

[MF30]

$\exp \mspace{11mu} \left( \frac{i\; 2\; \pi}{\mathcal{M}} \right)$

is difficult in the midst of phase errors and phase noise; but is animportant modulation for the sake of efficient use of radio resources(the number

[MF31]

log₂

-bit

is transmitted at once). It is, however, known that the realization ofthis modulation/demodulation is difficult.

To solve this problem, as shown in the lower part (the intermediateblock between the Switches 1-1 and 1-2) in FIG. 12, the transmitter

1) decomposes an integer (“information”) k given as

[MF32]

${k = {{j\; \mathcal{M}_{0}} + j^{\prime}}},{j = {\left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack \left( {= {{the}\mspace{14mu} {integer}\mspace{14mu} {part}\mspace{14mu} {of}\frac{k}{\mathcal{M}_{0}}}} \right)}}$${j^{\prime} = {\left\{ \frac{k}{\mathcal{M}_{0}} \right\} \left( {= {{the}\mspace{14mu} {fractional}\mspace{14mu} {part}\mspace{14mu} {of}\frac{k}{\mathcal{M}_{0}}}} \right)}};$

2) divides the delay-Doppler parameter space, called the target spaceequally into

[MF33]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

sub-parameter spaces and assigns 2-D PC (TD- and FD-PCs) to eachsub-space, where

[MF34]

(

≥

₀).

Furthermore, the transmitter 1) 2-D BPSK modulates a chip pulse by the

[MF35]

$i\mspace{11mu} \left( {,{0 \leq i \leq \left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack}} \right)$

-th 2-D PC; 2) combines it to form a

[MF36]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

-code-multiplexed signal;3) time-frequency-shifts the resultant signal (called a signature) aboutby around the center of the j-th sub-parameter space, i.e., by the timedelay

[MF37]

k _(d) ^((j))

and by the frequency delay

[MF38]

_(D) ^((j))

where

[MF39]

shift(k _(d) ^((j)),

_(D) ^((j))))

is referred to as the shift of the j-th Artificial Channel (AC);4) M₀-PSK modulates the time-frequency shifted signature by the j′-th

[MF40]

₀

-ary symbol

[MF41]

$\exp \mspace{11mu} \left( \frac{i\; 2\; \pi \; j^{\prime}}{\mathcal{M}_{0}} \right)\left( {,{0 \leq j^{\prime} \leq {\mathcal{M}_{0} - 1}}} \right)$

and transmits the modulated signal. Consequently, the

[MF42]

(k _(d) ^((j)),

_(D) ^((j))))

-shifted signature is again distorted by being passed through the MainChannel (MC) with shifts

[MF43]

(k _(d),

_(D)).

The inventor designs a CCF between an estimated and received templateand a received CE as follows. As shown in the middle block, connected tothe Switch2-1 in FIG. 13, The receiver1′) decomposes an estimate of k

[MF44]

{circumflex over (k)}

as

[MF45]

${\hat{k} = {{\hat{j\;}\mathcal{M}_{0}} + \hat{j^{\prime}}}},{\hat{j} = \left\lbrack \frac{\hat{k}}{\mathcal{M}_{0}} \right\rbrack},{{\hat{j^{\prime}} = \left\{ \frac{\hat{k}}{\mathcal{M}_{0}} \right\}};}$

2′) 2-D BPSK modulates a chip pulse by the

[MF46]

$\hat{j}\mspace{11mu} \left( {,{0 \leq \hat{j} \leq \left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack}} \right)$

-th 2-D PC;

3′) time-frequency-shifts it by the

[MF47]

ĵ

-th AC's [MF48]

shifts(k _(d) ^((ĵ)),

_(D) ^((ĵ))));

4′) and M₀-PSK demodulates the resultant signal by the

[MF49]

ĵ′

-th

[MF50]

₀

-ary symbol

[MF51]

$\exp \mspace{11mu} \left( \frac{i\; 2\pi \hat{\; j^{\prime}}}{\mathcal{M}_{0}} \right){\left( {,{0 \leq \hat{j^{\prime}} \leq {\mathcal{M}_{0} - 1}}} \right).}$

The resultant signal is an estimated and received template.

Maximization of the real parts of N′ TD-CCFs and N FD-CCFs using theirassociated LFs, in array forms, is performed in terms of the cardinalnumbers of the CCFs, as a function of the chip-level address (ρ′,ρ) andthe data-level address

[MF52]

{right arrow over (p)}=(p,p′),

and a pair of estimated decoding integers of k:

[MF53]

(ĵ,ĵ′).

The receiver 1) chooses the

[MF54]

ĵ

-th 2-D PC from

[MF55]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

2-D PCs; 2) cancels-out the phase of the data d_(p→) with address

[MF56]

{right arrow over (p)}=(p,p′)

by the PSK signal

[MF57]

${\exp \mspace{11mu} \left( \frac{{- i}\; 2\pi \hat{\; j^{\prime}}}{\mathcal{M}_{0}} \right)};$

3) maximizes the real parts of the 2 CCFs based the PUL algorithm, asshown in the lower block, connected to the Switch2-2, in FIG. 13, interms of

[MF58]

(k _(d),

_(D))

and

[MF59]

ĵ,ĵ′,

and 4) obtains the MLE of k

[MF60]

k*=j*

₀ +j* ^(,t).

This is realized by constructing a low

[MF61]

₀

-ary PSK-modulation-based encoder-decoder system and combining thesystem, together with exploiting

[MF62]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

ACs with non-commutative time-frequency shifts. This provides a high

[MF63]

-ary PSK modulation communication system. Namely, this system is capableof being used also for a synchronizer (or radar) for estimatingparameters, in cooperation with decoder of k, from the output signal ofone AC chosen, according to k, from

[MF64]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

ACs with non-commutative shifts, each of which is connected to the MCwith shifts

[MF65]

(k _(d),

_(D)).

Thus, the multiplexed system using non-commutative AC shifts may be apradigm shift. Note that the computational complexity of

[MF66]

-ary PSK-demodulation is about

[MF67]

$\left( {\sqrt{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} + \frac{\mathcal{M}_{0}}{\sqrt{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1}}} \right)$

times larger than that of synchronizer (or radar), O(N+N′):

<<Detailed Description of Theoretical Issues of the Patent and Examplesof Embodiment of Communication Systems>>

In accordance with an embodiment of the present invention, the inventordescribes details of theoretical issues of communication systems andspecific examples of an embodiment of the communication systems.

<1. Background of the Invention>

One of important problems to be solved in communication is to design awireless communication system that can accommodate more traffic within alimited amount of radio spectrum[1]. Orthogonal Frequency DivisionMultiplex (OFDM) has been the dominant technology, as a TD- andFD-division multiplexed system to convey data of time T_(s) andbandwidth F_(s) (see FIG. 1); but, it has shortcomings that theorthogonality is destroyed due to time and frequency offsets.

FIG. 1 shows three kinds of divisions of the time-frequency plane (TFP):(a) indicates time division, (b) frequency division, and (c) Gabordivision [1]. The solid lines in (a) show divisions of data of timeduration T; the thin lines subdivisions by Time Domain (TD)-Phase code(PC); The dotted lines in (b) show divisions of data of bandwidth F; thedashed lines subdivisions by Frequency Domain (FD)-PC.

“Synchronisation” is the first procedure for communication through thechannel with t_(d), f_(D). However, phase distortion (PD)s due to timeand frequency shift operator (TFSO) needed to the TD- and FD-divisionmultiplex

[MF68]

e ^(j2πT) ^(s) ^(F) ^(s)

are followed by the PD of the channel

[MF69]

e ^(j2πt) ^(d) ^(f) ^(D) .

Thus it is not easy to establish synchronisation. Moreover, there is noeffective solution to the problem of estimating t_(d), f_(D) from anecho signal of radars.

The inventor first modulates TD-signal and its FT, i.e., FD-signalsatisfying the time and frequency symmetrical property (TFSP) in termsof t_(d), f_(D) (see FIG. 2), by TD- and FD-PCs to design TD- andFD-signatures.

FIG. 2 shows an illustration of the NCP of TFSs (see [0014]), shows anillustration of the non-commutative property (NCP) of time and frequencyshift (TFS)s: The NCP is manifested in the expression of the product ofthe shift operators T_(τ,0)·T_(0,υ)=e^(−i2πτυ)T_(0,υ)T_(τ,0): Its LHScorresponds to the triangle in the figure; the RHS to the square; thePhase Distortion (PD) e^(−i2πτυ) appears; the circle in the figure showsthe Symmetrical Time-Frequency Shift Operator (TFSO)[26]

[MF70]

_(τ,υ)

e ^(iπτυ)

_(τ,0)·

_(0,υ).

Secondly, the inventor defines arrays of TD- and FD-CCFs (see FIG. 3b ),in the form of AFs[15] as optimum receivers for detecting templatesbased on the fact that PDs are embedded into signatures as templates,

As shown in [27]-[32], patent[1]-patent[6], the proposed estimationmethod firstly determines t_(d), f_(D) and the cardinal numbers of TD-and FD-CCFs maximizing the real parts of TD- and FD-CCFs; with noadvance information about t_(d), f_(D) and secondly updates estimates oft_(d), f_(D) alternatively. Furthermore, it is shown that this methodbecomes a good solution to the radar problem. FIG. 3a indicates theGabor function located on the TFP and its related issues: FIG. 3a 0shows Gaussian chip waveform g_(mm′)(t) located on the TFP and itsFourier Transform (FT) G_(mm′)(f); FIG. 3a 1 indicates the real andimaginary parts of the TD-template, the combination of g_(mm′)(t)weighted by the FD PC X′_(m′): FIG. 3a 2 indicates the real andimaginary parts of the FD-template, the combination of G_(mm′)(f)weighted by the TD PC X_(m); FIG. 3b shows NN′ Cross-CorrelationFunction (CCF)s, N′ column-sums of values of the TD-CCFs, and N row-sumsof values of the FD-CCFs; FIG. 3c shows the alternative projection,orthogonal projecting onto the Time-Limited Time Domain (TL-TD) and ontoBand-Limited Frequency Domain (BL-FD), based on the AlternativeProjection Theorem (APT), the estimated

[MF71]

{circumflex over (t)} _(d) ,{circumflex over (f)} _(D),

and convergent values t_(d), f_(D).

The problem of estimating t_(d), f_(D) from an echo signal is equivalentto that of determing two unknowns from PDs due to the NCP of TFSs.Hence, this belongs to the category of signal detection and estimationbased on the Weyl-Heisenberg Group (WHG) theory. However, exceptAuslander & Tolimieri's remark[17], many radar researchers haven'tobtained effective estimation methods with high precision yet becausethe NCP of TFSs was not taken into care in their estimation methods. Onthe contrary, the invention described in this disclosure is based on theinventor's belief that the NCP of TFSs only serves to strengthen theefficiency of communication systems including radars. The wavelettransform[8]. using the time and frequency shifted function, i.e., theGabor elementary functions

[MF72]

g _(m,m′)(t)

g(t−mT _(c))e ^(i2πm′F′) ^(c) ^((t−mT) ^(c) ⁾

discusses series representations of the form

[MF73]

${f(t)} = {\sum\limits_{m,m^{\prime}}{a_{m,m^{\prime}}e^{i\; 2\; \pi \; m^{\prime}v_{0}t}{g\left( {t - {m\; \tau_{0}}} \right)}}}$

and is primarily concerned with the coefficients a_(m,m′). In wirelesscommunication, the 5G, after 5G candidates: OFDM/OQAM, the filter bankmulti-carrier (FBMC), and GFDM [9, 10, 12] have an interest in designinga multiplexed signal f(t) with information a_(m,m′) to be transmittedand the elementary pulse g(t). Namely, the primary concern is the shapeof g(t) that makes intersymbol interference (ISI) and interchannelinterference (ICI) to zero and the orthogonality of g_(m,m′)(t).

In wireless communication, synchronisation that's resistant to time andfrequency offsets is necessary. However, there are a few attempts toestimate t_(d), f_(D). Furthermore, most of communication engineersconsiders that the PD due to TFSs mτ₀,m′ν₀

[MF74]

e ^(iπmr) ⁰ ^(m′υ) ⁰

can be negligible. However, the group theoretic property of the WHGtells us that the PD of the channel

[MF75]

e ^(iπt) ^(d) ^(f) ^(D)

is followed by the PD due to TFSs

[MF76]

e ^(iπmswm′υ) ⁰

and the PD due to the multi-carrier technique

[MF77]

e ^(iπm′v) ⁰ ^(t) ^(d)

arises simultaneously. The mechanism of phase errors arising is notsimple.

One must start by solving the following three themes behind the radarproblem. Note that Linear FM Continuous Wave with a chirp signal, acompressed pulse modulated by pulse of short time, and its multi-carrierversion are used as a transmit signal in conventional radars [19].

(theme1) The radar problem in range and velocity is essentially aproblem of two unknowns t_(d), f_(D); most of receivers is based onsearching peaks of the magnitude of complex-valued ambiguity function(AF) of two variables, called the ambiguity surface or using theAF-characteristic of a chirp signal, It's a natural belief that whensolving the problem relating to a given function of two unknowns, a wayof dealing jointly with (independent) some other functions of theunknowns may be a better solving method.

(theme2) The radar problem is suffering from the PD

[MF78]

e ^(j2πt) ^(d) ^(f) ^(D)

due to the NCP of the TFSs. Such a similar situation due to the NCP ofposition and momentum operators in quantum mechanics is observed. Achirp pulse sequence of time interval T_(p) and frequency shift F_(p)generates a PD

[MF79]

e ^(j2πT) ^(p) ^(F) ^(p) ,

Furthermore, a communication system multiplexed by non-overlappedsuperposition of a signal on the TFP to covey data-symbols of timeduration T_(s) and bandwidth F_(s), through the channel with t_(d),f_(D) is causing PD

[MF80]

e ^(j2πt) ^(d) ^(f) _(D)

followed by PD

[MF81]

e ^(j2πT) ^(s) ^(F) ^(s) .

(theme3) Closely relating to the theme2, it is not easy to understand amechanism whereby PD arises. That is, in communication systems andradars, two shift operators are usually defined as

[MF82]

S(a)f(t)=f(t+a),M(a)f(t)=e ^(i2πat) f(t), t∈

and give the NCP, represented as M(v)S(u)=e^(−i2πuv)S(u)M(v) i.e., theNCP is manifested as the PD e^(−i2πuv), in other words, the phase termof the product of two shifts comes from the fact that u, v areexponentiated. This observation is the same as the mathematical basisfor the introduction of the Heisenberg group in quantum mechanics,proposed by Weyl. Hence such PD is an important clue to solve the radaror synchronisation problem and is purely symbolic one. Carefulconsideration should be given to the exponential function, i.e.,e^(−i2πuv). This leads to the symmetrical TFSO (see FIG. 2 and (4),(24)) that is main concern in the present invention.

<2. Symmetrical Time-Frequency Shift Operators>

Let s(t) denote a real-valued pulse, and let σ(t) denote the Hilberttransform of s(t). Then one gets a complex signal, referred to as ananalytic signal, defined as ψ(t)=s(t)+iσ(t)[1]. A typical echo signalcan be represented in the form

[MF83]

r _(e)(t;t _(d) ,f _(D))=A

ψ(t−t _(d))e ^(iΩ(t−t) ^(d) ^()+iφ),  (1)

where ψ(t) is the Complex Envelope (CE) of a pulse,

[MF84]

A,t _(d),Ω,φ,Ω−Ω_(r)=2πf _(D)

denote its amplitude, time of arrival, carrier frequency, the phase ofits carrier, and a change in its carrier frequency, called the Dopplershift of the reference carrier Ω_(r)=2πf_(r). For brevity, let us assumeΩ_(r)=0 for a moment. Let

[MF85]

[⋅]

denote the Fourier Transform (FT) and

[MF86]

Ψ(f)=

[ψ(t)]

denote the FT of ψ(t), then the FT of r_(e)(t; t_(d), f_(D)) is in theform

[MF87]

R _(e)(f;t _(d) f _(D))=A

[ψ(t−t _(d))e ^(i2πf) ^(D) ^((t−t) ^(d) ⁾]e ^(iφ) =A

Ψ(f−f _(D))e ^(−i2πft) ^(d) ^(+iφ).  (2)

A pair of r_(e)(t; t_(d), f_(D)) and R_(e)(f; t_(d), f_(D)) is notsymmetrical in t_(d) and f_(D) because the product of unknowns t_(d) andf_(D) appears only in the TD function (1).

But it can be represented in a slightly modified form[24, 26]

[MF88]

$\begin{matrix}{{\mathcal{F}\left\lbrack {{\psi \left( {t - t_{d}} \right)}e^{i\; 2\; \pi \; {f_{D}{({t - \frac{t_{d}}{2}})}}}} \right\rbrack} = {{\Psi \left( {f - f_{D}} \right)}{e^{{- i}\; 2\pi \; {t_{d}{({f - \frac{f_{D}}{2}})}}}.}}} & (3)\end{matrix}$

If a TD function x(t) and its FD function

[MF89]

X(f)=

[x(t)]

are symmetrical in terms of t_(d) and f_(D), then this property iscalled the time and frequency symmetrical property (TFSP) (see FIG. 2).Thus one can define a symmetrical time-frequency shift operators (TFSO)satisfying the TFSP, given as

[MF90]

$\begin{matrix}\left. \begin{matrix}{\left. {{_{t_{d},f_{D}}{x(t)}} = {x\left( {t - t_{d}} \right)}} \right)e^{{i\; 2\pi \; {f_{D}{({t - \frac{t_{d}}{2}})}}},}} \\{{_{f_{D} - t_{d}}^{f}{X(f)}} = {{X\left( {f - f_{D}} \right)}e^{{- i}\; 2\pi \; {t_{d}{({f - \frac{f_{D}}{2}})}}}}}\end{matrix} \right\} & (4)\end{matrix}$

and get a trivial identity between the two operators

(Property 1 of the Symmetrical TFSOs): [MF91]

_(f) _(D) _(,−t) _(d) ¹=

_(d,f) _(D)

⁻¹.  (5)

Usually, the time shift operator S(−t_(d))x(t)=x(t−t_(d)) and themodulation operator

[MF92]

M(f _(D))x(t)=e ^(i2πf) ^(D) ^(t) x(t)

are used. On the contrary, the shifts

[MF93]

$\frac{t_{d}}{2}$

in a TD signal x(t) and

[MF94]

$\frac{f_{D}}{2}$

in its FD signal X(f) in (4), hereafter called “half shifts”, seem to bea few slight modifications to usual time-frequency representations ofsignal:

[MF95]

S(−t _(d))M(f _(D))x(t)=x(t−t _(d))e ^(i2πf) ^(D) ^((t−t) ^(d) ⁾

or

[MF96]

M(f _(D))S(−t _(d))x(t)=x(t−t _(d))e ^(i2πf) ^(D) ^(t).

But such half shifts are our intention to get effective representationsof a radar signal and its received one so that their phase factors arefully traced in both the TD and FD as shown below.

Proposition 1: The TFSO (4) is identical to von Neumann's canonicalcommutative relations(usually abbreviated to CCRs) in quantum mechanics[4, 6] as referred to as the Stone-von Neumann theorem [5, 4, 25],defined as a two-parameter family of unitary operators S(a,b)=e^(−1/2 iab)U(a)V(b) with its group-theoretic property, where U(a),V(b) are defined below. T_(τ,ν) and T^(f) _(ν,−τ) are referred to as vonNeumann's TFSO s hereafter.

The Heisenberg commutation relation, referred to as the Heisenberg'suncertainty principle, is given as[4, 5]

[MF97]

[Q,P]=QP−PQ=ih.  (6)

Proposition 2: Let us assume that Q and P can be exponentiated toone-parameter unitary groups U(a)=exp(iaQ), V(b)=exp(ibP), respectively(a, b real) and let us associate Q, P with TFSOs T_(τ,0), T_(0,ν),respectively, then

[MF98]

$\begin{matrix}{\left\lbrack {_{\tau,0},_{0,v}} \right\rbrack = {{- 2}i\mspace{14mu} {\sin \left( {\pi {\begin{matrix}\tau & 0 \\0 & v\end{matrix}}} \right)}{_{\tau,v}.}}} & (7)\end{matrix}$

Thus, the classical limit, i.e.,

[MF99]

h→0

limit[20] corresponds to

[MF100]

[

_(r,0),

_(0,ν)]=0, PD-free, i.e., τν∈

,  (8)

where

[MF101]

h _(s)[⋅,⋅]

denote the reduced Planck's constant

[MF102]

$\frac{h}{2\pi}$

and the quantum mechanics commutator.

The composition of TFSOs

[MF103]

_(t₁, f₁)_(t₂, f₂) = e^(i π(t₁f₁ + t₂f₂))_(t₁, 0)_(0, f₁)_(t₂, 0)_(0, f₂) = e^(i π(t₁f₁ + t₂f₂ + 2t₂f₁))_(t₁, 0)_(t₂, 0)_(0, f₁)_(0, f₂) = e^(i π(t₁f₁ + t₂f₂ + 2t₂f₁ − (t₁ + t₂)(f₁ + f₂)))_(t₁ + t₂, f₁ + f₂)

gives

(Property 2 of the Symmetrical TFSOs): [MF104]

$\begin{matrix}\left. \begin{matrix}{{{_{t_{1},f_{1}}_{t_{2},f_{2}}} = {e^{{- i}\; \pi {\begin{matrix}t_{1} & f_{1} \\t_{2} & f_{2}\end{matrix}}}_{{t_{1} + t_{2}},{f_{1} + f_{2}}}}},} \\{{{_{f_{1},{- t_{1}}}^{f}_{f_{2},{- t_{2}}}^{f}} = {e^{{- i}\; \pi {\begin{matrix}t_{1} & f_{1} \\t_{2} & f_{2}\end{matrix}}}_{{f_{1} + f_{2}},{{- t_{1}} - t_{2}}}^{f}}},}\end{matrix} \right\} & (9)\end{matrix}$

Eq. (7) is an example of the first equation in eq. (9).

The fact that the product of unknowns t_(d) and f_(D) appearsymmetrically in the exponent of PDs of TD- and FD-functions from theirsymmetry property is the more important property of the symmetricalTFSOs. Chip- and data-level addresses of the multiplexed signal appearedin the the exponent of PDs as well. The product of such exponentsenables us to easily estimate parameters as discussed below. Forexample, in wireless communication[9, 10] an orthogonal frequencydivision multiplex (OFDM) signal

[MF105]

$\begin{matrix}{{{f(t)} = {\sum\limits_{m,n}{a_{m,n}e^{i\; 2\pi \; {mv}_{0}t}{x\left( {t - {n\; \tau_{0}}} \right)}}}},{{{with}\mspace{14mu} v_{0}\tau_{0}} = 1}} & (10)\end{matrix}$

is the main subject, where the coefficients a_(m,n) take complex valuesrepresenting an encoded transmit data, and x(t) is a prototype function.The OFDM signal is rewritten as

[MF106]

$\begin{matrix}{{f(t)} = {\sum\limits_{m,n}{a_{m,n}e^{i\; \pi \; n\; \tau_{0}{mv}_{0}}_{{n\; \tau_{0}},{mv}_{0}}{{x(t)}.}}}} & (11)\end{matrix}$

Hence the PD e^(iπnτ0mν0) does not alter the a_(m,n) except its sign asfar as the condition τ₀ν₀=1 holds (cf. (8)). But if there are offsetssuch as τ′=τ₀+ε_(τ) and ν′=ν₀+ε_(ν), then the PD

[MF107]

$e^{{- i}\; \pi {\begin{matrix}{\tau_{0} + ɛ_{\tau}} & 0 \\0 & {v_{0} + ɛ_{v}}\end{matrix}}} = {\left( {- 1} \right)^{\tau_{0}v_{0}}e^{{- i}\; {\pi {({{ɛ \cdot v_{0}} + {ɛ_{v} \cdot \tau_{0}} + {ɛ_{\tau}ɛ_{v}}})}}}}$

inevitably arises followed by the PD

[MF108]

e ^(−iπt) ^(d) ^(f) ^(D)

through a doubly dispersive channel with t_(d) and f_(D). That is, wehave to confront a phase-distorted signal

[MF109]

$\begin{matrix}{{_{t_{d},f_{D}}{f(t)}} = {\sum\limits_{m,n}{a_{m,n}e^{{- i}\; {\pi {({{t_{d}f_{D}} + {2t_{d}{mv}_{0}}})}}}{x\left( {t - {n\; \tau_{0}} - t_{d}} \right)}{e^{i\; 2{\pi {({{mv}_{0} + f_{D}})}}t}.}}}} & (12)\end{matrix}$

The conventional non-overlapping superposition of a signal in the TFP[1, 2, 9, 10, 12], defined as in (10) is causing an accumulation of thePD

[MF110]

e ^(iπ2t) ^(d) ^(mν) ⁰

because of its group-theoretic property like (12). Hence the PDsdirectly weaken the output of the receiver and give a seriousdeterioration in “synchronisation” needed in digital communicationsystems[3]. On the contrary, the main concern is the design of theprototype function x(t) to reduce inter-symbol interference (ISI) andinter-channel interference (ICI). Such simple observation provides astarting point for this study.

<3. Likelihood Functionals and Cross-Correlation Functions>

To begin with, we have to study a brief review of Woodward's[15]statistical approach to the analysis and design of optimum radarsystems, and Helstrom's[18] comprehensive study on theory and practiceof signal detection and estimation.

It is an important observation that these approaches used only Abelianharmonic analysis as Auslander and Tolimieri's remark on Wilcox'sstudy[16].

However, the fundamentals of radar theory reside in the following signaldetection and parameter estimation theory.

<3.1 Signal Detection>

When a radar signal appears in a receiver, its detection is madeuncertain by the simultaneous presence of noise.

Consider the simplest signal-detection problem, that of deciding whethera signal s(t) of specified form has arrived at a definite time in themidst of Gaussian noise n(t). An input w(t) to the receiver is measuredduring an observation interval

[MF111]

0≤t≤T.

On the basis of this input an observer must choose one of twohypotheses, H₀, “there is no signal,” i.e., w(t)=n(t), and H₁, “thesignal is present,” i.e., w(t)=s(t)+n(t). When w_(k)=w(t_(k)) ismeasured at time t=t_(k) during the observation interval, the n samplesw_(k) are random variables having a joint probability density function(p.d.f.) p_(i)(w) under hypotheses H_(i), i=0, 1, and the observer'sdecision is best made on the basis of the likelihood ratioΛ(w)=p₁(w)/p₀(w), w=(w₁, . . . , w_(n)).

For a fixed decision level Λ₀ the observer chooses hypothesis H₀ ifΛ(w)<Λ₀; H₁ if Λ(w)>Λ₀.

A radar signal can be written simply as[15, 18]

[MF112]

s(t)=

ψ(t)e ^(iΩt),

where ψ(t) is its CE and Ω=2πf_(c) the carrier frequency. If thespectrum of the signal s(t)

[MF113]

S(f)=

[s(t)]=½[Ψ(f−f _(c))+Ψ*(−f−f _(c))], Ψ(f)=

[ψ(t)]

exhibits two narrow peaks, one near the frequency f_(c) and the othernear −f_(c), and if the widths of the bands are much smaller than Ω, thesignal is termed narrowband (NB) or quasi-harmonic.

Assume that the input to the receiver

[MF114]

w(t)=

ψ_(w)(t)e ^(iΩt)

is NB and the CE [MF115]

ψ_(w)(t)

can be measured by a mixer.

In the presence of stationary NB white Gaussian noise with theauto-covariance function

[MF116]

ϕ(τ)=

{tilde over (ϕ)}(τ)e ^(iΩτ),{tilde over (ϕ)}(τ)=N ₀δ(τ),

an optimum detector of the NB signal

[MF117]

s(t)=

ψ(t)e ^(i(Ωt+φ))

has the logarithm of its LF (LLF)[18, p. 106]

[MF118]

$\begin{matrix}{{{\ln \; {\Lambda \left\lbrack {\psi_{w}(t)} \right\rbrack}} = {g - \frac{d^{2}}{2}}},{g = {\frac{e^{{- i}\; \phi}}{N_{0}}{\int_{0}^{T}{{\psi^{*}(t)}{\psi_{w}(t)}{dt}}}}},{d^{2} = {\frac{1}{N_{0}}{\int_{0}^{T}{{{\psi (t)}}^{2}{dt}}}}},} & (13)\end{matrix}$

in which N₀ is the unilateral spectral density of the white noise, g,being generated by passing the input ψ_(w)(t) through a filter[19]matched to the signal ψ(t) to be detected, and d² are referred to as thestatistic and the signal-to-noise ratio (SNR) of the LF Λ[ψ_(w)(t)].

<3.2 Estimation of Signal Parameters>

The principle in hypothesis testing can be applied to choices amongmultiple hypotheses as follows. Suppose that a transmitter is sending asignal using one of M signals. The receiver is to decide which of theseM signals is present during the observation interval (0,T). Namely,under hypothesis H_(k), “a signal s_(k)(t) was sent”, an input to thereceiver is

[MF119]

w(t)=s _(k)(t)+n(t),s _(k)(t)=

ψ_(k)(t)e ^(i(2πf) ^(k) ^(t+φ) ^(k) ⁾, 1≤k≤M,  (14)

where ψ_(k)(t) is the NB CE, f_(k) the carrier,

[MF120]

φ_(k)

denotes the phase of s_(k)(t), and n(t) random noise.

The receiver chooses one of the M hypotheses on the basis ofmeasurements of its input w(t). Suppose that the receiver makes nmeasurements w₁, . . . , w_(n) of its input w(t). Let p_(k)(w) be thejoint p.d.f. of these data under hypothesis H_(k) and let ζ_(k) be theprior probability of that hypothesis. The likelihood ratio for detectingthe k th signal in the presence of n(t) is defined byΛ_(k)(w)=p_(k)(w)/p₀(w), where p₀(w) denotes the p.d.f. under a dummyhypothesis. For simplicity, assume that ζ_(k)=M⁻¹, under theorthogonality of the signals s_(k)(t)

[MF121]

∫₀ ^(T) s* _(i)(t)s _(j) dt=E _(i)δ_(ij),

(E_(i) is the energy of the i th signal) then the receiver simplydecides H_(k) if Λ_(k)(w)>Λ_(j)(w), for all k≠j.

Denote the unknown parameters of the signal by θ₁, . . . , θ_(m) andrepresent them by a vector θ=(θ₁, . . . , θ_(m)) in an m-dimensionalparameter space, designated by Θ.

A radar echo can be represented in the form

[MF122]

s _(echo)(t;A,κ,t _(d) ,f _(D))=A

e ^(iκ)ψ(t−t _(d))e ^(i2πf) ^(c) ^((t−t) ^(d) ⁾ , f _(c) =f _(r) +f_(D),  (15)

where Ae^(iκ) denotes the attenuation factor, A its amplitude, t_(d) itstime of arrival, f_(c) its carrier frequency, κ the phase of itscarrier, and f_(D) the Doppler shift of its reference carrier f_(r).Unknown parameters in the echo signal (15) are given as θ=(A, κ, t_(d),f_(D)). When

[MF123]

w(t)=

ψ_(w)(t)e ^(i2πf) ^(r) ^(t)

and the noise is white with unilateral spectral density N₀, the LLF[18,p. 251] is

[MF124]

${\ln \; {\Lambda \left\lbrack {{w(t)};\theta} \right\rbrack}} = {{\left\lbrack {\frac{{Ae}^{{- i}\; \kappa}}{N_{0}}{\int_{0}^{T}{{\psi^{*}\left( {t - t_{d}} \right)}e^{{- i}\; 2\pi \; {f_{D}{({t - t_{d}})}}}{\psi_{w}(t)}{dt}}}} \right\rbrack} - {\frac{A^{2}}{2N_{0}}{\int_{0}^{T}{{{\psi \left( {t - t_{d}} \right)}}^{2}{{dt}.}}}}}$

Using a change of variables such as

[MF125]

Ae ^(iκ) =u+iv, u=

Ae ^(iκ) , v=Ae ^(iκ).

Here [MF126]

ℑ

denotes the imaginary part of the complex number following it. Thus onegets the maximum likelihood estimate (MLE) of A, κ containing θ′=(t_(d),f_(D)), given by [18, p. 251].

[MF127]

$\begin{matrix}\left. \begin{matrix}{{{{\hat{A}\left( \theta^{\prime} \right)}e^{i\; {\hat{\kappa}{(\theta^{\prime})}}}} = {{\hat{u} + {i\hat{v}}} = \frac{z\left( \theta^{\prime} \right)}{d^{2}\left( \theta^{\prime} \right)}}},{\hat{u} = {{\hat{A}\left( \theta^{\prime} \right)}^{i\; {\hat{\kappa}{(\theta^{\prime})}}}}},{\hat{v} = {\; {\hat{A}\left( \theta^{\prime} \right)}^{i\; {\hat{\kappa}{(\theta^{\prime})}}}}},} \\\begin{matrix}{{{z\left( \theta^{\prime} \right)} = {\frac{1}{N_{0}}{\int_{0}^{T}{{\psi^{*}\left( {t - t_{d}} \right)}e^{{- i}\; 2\pi \; {f_{D}{({t - t_{d}})}}}{\psi_{w}(t)}{dt}}}}},} \\{{d^{2}\left( \theta^{\prime} \right)} = {\frac{1}{N_{0}}{\int_{0}^{T}{{{\psi \left( {t - t_{d}} \right)}}^{2}{{dt}.}}}}}\end{matrix}\end{matrix} \right\} & (16)\end{matrix}$

The MLEs of the remaining parameters θ′ are those that maximize[18, p.251]

[MF128]

${\max\limits_{u,v}{\ln \; {\Lambda \left\lbrack {\left. {w(t)} \middle| u \right.,v,\theta^{\prime}} \right\rbrack}}} = {\frac{{{z\left( \theta^{\prime} \right)}}^{2}}{2{d^{2}\left( \theta^{\prime} \right)}}.}$

Hence one can concentrate one's efforts on estimating the θ′. For one ofa closely spaced set of values of the Doppler shift

[MF129]

${f_{D} \in \left( {{- \frac{W_{D}}{2}},\frac{W_{D}}{2}} \right)},$

the MLE θ′ could be obtained by building a bank of parallel filters,each matched to a signal of the form

[MF130]

ψ(t)e ^(i2π(f) ^(r) ^(+f) ^(D) ^()t),

where W_(D) is the maximum range of the expected Doppler shift. It is,however, not easy to examine statistics by constructing a bank of NBfilters in parallel. This fact leads one to decompose the2-unknown-parameter problem into 2 single-unknown-parameter problems.

Let us rewrite (15) and its FT in the form

[MF131]

s _(echo)(t;A,κ,t _(d) ,f _(D))=A

e ^(iκ)

_(t) _(d) _(,f) _(D)

_(0,f) _(c) ψ(t)e ^(iφ) ⁰ ,

S _(echo)(f;A,κ,t _(d) ,f _(D))=A

e ^(iκ)

_(f) _(D) _(,−t) _(d) ^(f)

_(f) _(r) _(,0) ^(f)Ψ(f)e ^(iφ) ⁰ ,Ψ(f)=

[ψ(t)],

where

[MF132]

_(t) _(d) _(,f) _(D)

is a von Neumann's TFSO with t_(d) and f_(D) where

[MF133]

ψ(t),φ₀

denote the CE to be designed and its phase (its detail is omitted here),and

[MF134]

_(0,f) _(r) φ(t)

a passband signal shifted from the baseband CE by the referencefrequency f_(r).

Using pulse code techniques with the TD-PC of period N and the FD-PC ofperiod N′, one can want to make clear the exact location of (t_(d),f_(D)) in a 2-dimensional lattice

[MF135]

T _(c)

×F _(c)

in the TFP, where T_(c)=T_(s)/N, F_(c)=F_(s)/N′, T_(s), and F_(s) denotea chip-pulse spacing, chip (sub-) carrier-spacing, signal (ordata)-duration, and carrier-spacing, Divide the (t_(d), f_(D))-parameterspace Θ′ into a large number NN′ of small rectangular regions Δ^(q→)_(m,m′) with the data address

[MF136]

{right arrow over (q)}=(q,q′),

defined as

[MF137]

$\begin{matrix}{{\Delta_{m,m^{\prime}}^{\overset{\rightarrow}{q}} = \left\{ {\left. {\left( {t_{d},f_{D}} \right) \in {\mathbb{R}}^{2}} \middle| {{mT}_{c} \leq {t_{d} - {qT}_{s}} < {\left( {m + 1} \right)T_{c}}} \right.,{{m^{\prime}F_{c}} \leq {f_{D} - {q^{\prime}F_{s}}} < {\left( {m^{\prime} + 1} \right)F_{c}}}} \right\}},\mspace{20mu} {0 \leq m \leq {N - 1}},{0 \leq m^{\prime} \leq {N^{\prime} - 1}}} & (17)\end{matrix}$

and denote by H_(m,m′) the proposition “The parameter set θ′ lies inregion

[MF138]

Δ_(m,m′) ^({right arrow over (q)}).”

However, NN′ such hypotheses H_(m,m′) can be decomposed into N′hypotheses for estimating f_(D) in the TD signal s_(echo)(t; A, κ,t_(d), f_(D)) and N hypotheses for estimating t_(d) in the FD signalS_(echo)(f; A, κ, t_(d), f_(D)) as discussed below.

Provided that the signals s_(k)(t;θ′) are orthogonal, in hypothesisH_(k) associated with (14), for the CE ψ_(k)(t) and phase

[MF139]

φ_(k),

consider the kth NB echo signal

[MF140]

s _(k)(t;θ′)=A

e ^(iκ)ψ_(k)(t−t _(d))e ^(i2π(f) ^(k) ^(+f) ^(D) ^()(t−t) ^(d) ^()+iφ)^(k) .

If the noise is white and Gaussian with unilateral spectral density N₀,then its TD-LLF [18, p. 129, p. 251] is

[MF141]

$\begin{matrix}{\mspace{20mu} {{{\ln \; {\Lambda_{k}\left\lbrack {{{w(t)};A},\kappa,\theta^{\prime}} \right\rbrack}} = {{g_{k}\left( \theta^{\prime} \right)} - \frac{d_{k}^{2}\left( \theta^{\prime} \right)}{2}}},\mspace{20mu} {{d_{k}^{2}\left( \theta^{\prime} \right)} = {\frac{A^{2}}{N_{0}}{\int_{0}^{T}{{{\psi_{k}\left( {t - t_{d}} \right)}}^{2}{dt}}}}},{{g_{k}\left( \theta^{\prime} \right)} = {\; \frac{{Ae}^{- {i{({\kappa + \phi_{k}})}}}}{N_{0}}{\int_{0}^{T}{{\psi_{k}^{*}\left( {t - t_{d}} \right)}e^{{- i}\; 2{\pi {({f_{k} + f_{D} - f_{w}})}}{({t - t_{d}})}}{\psi_{w}(t)}{{dt}.}}}}}}} & (18)\end{matrix}$

Let k=k₀ be an integer satisfying

[MF142]

${\max\limits_{1 \leq k \leq M}\frac{{g_{k}\left( \theta^{\prime} \right)}}{d_{k}\left( \theta^{\prime} \right)}} > r_{0}$

for a given decision level r₀; Then the receiver decides the kth signalhas been arrived; If all statistics g_(k) lie below r₀, the receiverdecides that no signal was transmitted. This is referred to as an MLreceiver. Hence a construction of orthogonal signals s_(k)(t) is ofcrucial importance. Equation (18) suggests two ways of maximizing|g_(k)(θ′)|: one is maximizing the integrand and the other is cancelingout the phase factors e^(iκ) of the carrier and

[MF143]

e ^(iφ) ^(k)

of the signal s_(k)(t). The phase factor, however, is usually absorbedinto

[MF144]

ψ_(k)(t)

by re-definition; moreover not g_(k)(θ′) but |g_(k)(θ′)| is simplyevaluated. These strategies annihilate the phase information.Woodward[15] used a two-dimensional CCF, called the ambiguity function(AF), early defined by Ville[14], that plays a central role in the radarsignal design. It is given by the definition

[MF145]

${{\theta_{xy}\left( {\tau,v} \right)} = {\int_{- \infty}^{\infty}{{x\left( {t + \frac{\tau}{2}} \right)}{y^{*}\left( {t - \frac{\tau}{2}} \right)}e^{{- i}\; 2\pi \; {vt}}{dt}}}},{{\Theta_{XY}\left( {v,{- \tau}} \right)} = {\int_{- \infty}^{\infty}{{X\left( {f + \frac{v}{2}} \right)}{Y^{*}\left( {f - \frac{v}{2}} \right)}e^{i\; 2\pi \; f\; \tau}{df}}}},{{X(f)} = {\mathcal{F}\left\lbrack {x(t)} \right\rbrack}},{{Y(f)} = {{\mathcal{F}\left\lbrack {y(t)} \right\rbrack}.}}$

Non-commutative and group-theoretic properties of WHG-basedtime-frequency-shifted waveforms are manifested in the cisoidal factorsin (9).In addition, one can get:Proposition 3 [Property3 of symmetric TFSOs]: For a TD signal z(t) andits FT

[MF146]

Z(f)=

[z(t)],

the inner product (IP)s between the TD and FD time-frequency-shiftedsignals can be expressed as

[MF147]

$\begin{matrix}\left. \begin{matrix}{{{\langle{{_{t_{1},f_{1}}{z(t)}},{_{t_{2},f_{2}}{z(t)}}}\rangle}_{t} = {e^{{- i}\; \pi {\begin{matrix}t_{1} & f_{1} \\t_{2} & f_{2}\end{matrix}}}\theta_{z,z}\left( {{t_{2} - t_{1}},{f_{2} - f_{1}}} \right)}},} \\{{{\langle{{_{f_{1},{- t_{1}}}^{f}{Z(f)}},{_{f_{2},{- t_{2}}}^{f}{Z(f)}}}\rangle}_{f} = {e^{{- i}\; \pi {\begin{matrix}t_{1} & f_{1} \\t_{2} & f_{2}\end{matrix}}}{\Theta_{Z,Z}\left( {{f_{2} - f_{1}},{{- t_{2}} + t_{1}}} \right)}}},}\end{matrix} \right\} & (19)\end{matrix}$

where

[MF148]

<r(t),s(t)>_(t)=∫_(−∞) ^(∞) r(t)s*(t)dt

denotes the TD IP of r(t) and s(t), and the FD IP

[MF149]

<R(f),S(f)>_(f)=∫_(−∞) ^(∞) R(f)S*(f)df=<r(t),s(t)>_(t)

of

[MF150]

R(f)=

[r(t)]

and

[MF151]

S(f)=

[s(t)].

Equation (19) shows that: i) both the real parts of the TD and FD IPsare maximized when t₂=t₁ and f₂=f₁, for which the maxima of the AF areattained; ii) if the left and right terms of the IP are thought of asthe input signal ψ_(w)(t) and a signal ψ(t) to be detected, respectivelyof the statistic g in LLF (13), then ψ(t) may cancel-out the inputsignal's PD to enhance the statistic g. Proposition 3 emphasizes theimportance of “phase” of a modulated signal like the “phasors” used inanalysing alternating currents and voltages in electricalengineering[18, xv, p. 91], and tells us that two quantities t_(d) andf_(D) always appear in a phase term. This is a major step toward (t_(d),f_(D))-estimations in the TD and FD, based on the WHG theory, indistinct contrast to the conventional methods using matched filters[18],performed only in the TD. If one can design a signal so that its phaseterms have been endowed with easy-traceability, one may utilize the boththe PDs of ψ_(w)(t), ψ(t) of the statistic in the TD- and FD-LLFs.

Gabor[1] stressed the importance of analysis in the TFP and the utilityof the Gaussian wave 2^(1/4)e^(−πt{circumflex over ( )}2) attaining thelower bound of the uncertainty relation of time and frequency.

He gave the time-frequency representation of a function f:

[MF152]

$\begin{matrix}\left. \begin{matrix}{{{f(t)} = {\sum\limits_{n,{m \in {\mathbb{Z}}}}{a_{m,n}{g_{m,n}(t)}}}},{a_{m,n} = {\langle{g_{m,n},f}\rangle}_{t}},m,{n \in {\mathbb{Z}}},} \\{{{g_{m,n}(t)} = {{g\left( {t - {n\; \tau_{0}}} \right)}e^{2\pi \; {imv}_{0}t}}},{{\tau_{0}v_{0}} = 1},m,{n \in {{\mathbb{Z}}.}}}\end{matrix} \right\} & (20)\end{matrix}$

It is well known [9, p. 985] that the set of Gaussian functionsg_(m,n)(t) forms a basis of L²(R) having good properties regarding timeand frequency localization but this basis is not orthonormal and the setof the functions is not even a frame[7]. See [7, 9] for a review on thedouble series representation (20). Most of communication engineers doesnot employ a Gaussian function primarily because it does not satisfy theNyquist condition. In our (t_(d), f_(D))-estimation problem, however,several favourable properties of Gaussians in the TFP play an importantrole.

<4. Signature Waveforms and Templates in TD and FD>

TD-PC techniques, i.e., spreading spectrum (SS) techniques[3] canprovide the simultaneous use of a wide frequency band, via code-divisionmultiple access (CDMA) techniques, in which a signal to be transmitteds(t) is modulated by an independent pulse code c(t) so that itsbandwidth is much greater than that of the message signal m(t), e.g.,s(t)=m(t)c(t) and each user is assigned a pulse code such that thesignals are orthogonal.

In order for the CE

[MF153]

ψ(t)

to satisfy the time and frequency symmetrical property (TFSP), its FTΨ(f) should also be phase coded.

In place of continuous-time signals in

[MF154]

₂(

),

consider discrete-time signals in

[MF155]

₂(

)

and assume that a TD signal s(t) is sampled with a sampling interval Δt,while a discrete-frequency signal is obtained by the L-point discreteFourier transform (DFT). Hence the frequency gap between two adjacentfrequency bins in the FD is Δf=1/(LΔt).

Let

[MF156]

$k = \left\lfloor \frac{t}{\Delta \; t} \right\rfloor$

(the truncation of fraction

[MF157]

$\left. \frac{t}{\Delta \; t} \right)$

and

[MF158]

$ = \left\lfloor \frac{f}{\Delta \; f} \right\rfloor$

(the truncation of fraction

[MF159]

$\left. \frac{f}{\Delta \; f} \right)$

be discrete variables of time t and frequency f. For the orthogonalityof a chip pulse, let assume

[MF160]

$F_{c} = \frac{1}{T_{c}}$

and let

[MF161]

T _(c) =MΔt,F _(c) =M′Δf.

Thus one can define the L-point twiddle factor

[MF162]

${W = e^{- \frac{i\; 2\pi}{L}}},{L = {M\; {M^{\prime}.}}}$

Define the following 7 different kinds of discrete-time anddiscrete-frequency signals:

[MF163]

-   -   TD, FD pulse waveforms: g[k], G[        ],    -   TD, FD templates: u_(m′) ⁽³⁾[k; X], U_(m) ⁽⁴⁾[        ; X′],    -   TD, FD signatures: v[k; χ], V[        ; χ],    -   TD, FD transmit signals: s[k; χ], S[        ; χ],    -   CE of s[k; χ], FT of ψ[k; χ]:ψ[k; χ], Ψ[        ; χ],    -   TD, FD received signals: r[k; χ], R[        ; χ],    -   CE of r[k; χ], FT of ψ_(r)[k; χ]:ψ_(r)[k; χ], Ψ_(R)[        ; χ],        where X=(X₀, . . . , X_(N−1))∈{−1,1}^(N) is a TD-PC of period N,        X′=(X′₀, . . . , X′_(N′−1))ϵ{−1,1}^(N′) an FD-PC of period N′,        and χ=(X, X′).

For a continuous chip-pulse g(t) having support [−LΔt/2,LΔt/2], i.e.,duration LΔt, one can obtain its causal discrete-time LΔt-time-limited(TL) chip-pulse g[k] with delay[10] (D/2)Δt,D=L−1, L=(ΔtΔf)⁻¹=MM′

[MF164]

$\begin{matrix}{{{g\lbrack k\rbrack} = {\sqrt{\Delta \; t}{g\left( {\left( {k - \frac{D}{2}} \right)\Delta \; t} \right)}}},{{k} \leq {L/2}}} & (21)\end{matrix}$

and define its discrete-frequency LΔf-band-limited (BL) chip-pulse G[≃]having support [−LΔf/2,LΔf/2], i.e., bandwidth LΔf, by the DFT of g[k]

[MF165]

${{G\lbrack \rbrack} = {{\mathcal{F}^{d}\left\lbrack {g\lbrack k\rbrack} \right\rbrack} = {\frac{1}{\sqrt{L}}{\sum\limits_{k = 0}^{L - 1}{{g\lbrack k\rbrack}W^{k\; }}}}}},{0 \leq  \leq {L - 1.}}$

Introduce now a discrete-time TD-signature v[k;χ] and an FD-signature V[

;χ] defined as

[MF166]

$\begin{matrix}\left. \begin{matrix}{{{v\left\lbrack {k;} \right\rbrack} = {\frac{1}{\sqrt{N^{\prime}}}{\sum\limits_{m^{\prime} = 0}^{N^{\prime} - 1}{X_{m^{\prime}}^{\prime}_{0,{m^{\prime}M^{\prime}}}^{d}{u_{m^{\prime}}^{(3)}\left\lbrack {k;X} \right\rbrack}}}}},} \\{{{V\left\lbrack {;} \right\rbrack} = {\frac{1}{\sqrt{N}}{\sum\limits_{m = 0}^{N - 1}{X_{m}_{0,{- {mM}}}^{f,d}{U_{m}^{(4)}\left\lbrack {;X^{\prime}} \right\rbrack}}}}},}\end{matrix} \right\} & (22)\end{matrix}$

in terms of a TD-template of type-3

[MF167]

u _(m′) ⁽³⁾[k;X]

and an FD-template of type-4

[MF168]

U _(m) ⁽⁴⁾[

;X′],

respectively defined by

$\begin{matrix}\left. \begin{matrix}\begin{matrix}{{{u_{m^{\prime}}^{(3)}\left\lbrack {k;} \right\rbrack} = {\frac{1}{\sqrt{N}}{\sum\limits_{m = 0}^{N - 1}{X_{m}e^{{- i}\; \pi \; m\; m^{\prime}{MM}^{\prime}\Delta \; t\; \Delta \mspace{11mu} t}_{{mM},0}^{d}{g\lbrack k\rbrack}}}}},} \\{{0 \leq m^{\prime} \leq {N^{\prime} - 1}},}\end{matrix} \\\begin{matrix}{{{U_{m}^{(4)}\left\lbrack {;^{\prime}} \right\rbrack} = {\frac{1}{\sqrt{N^{\prime}}}{\sum\limits_{m^{\prime} = 0}^{N^{\prime} - 1}{X_{m^{\prime}}^{\prime}e^{i\; \pi \; m\; m^{\prime}{MM}^{\prime}\Delta \; t\; \Delta \mspace{11mu} t}_{{m^{\prime}M^{\prime}},0}^{f,d}{G\lbrack \rbrack}}}}},} \\{{0 \leq m \leq {N - 1}},}\end{matrix}\end{matrix} \right\} & (23)\end{matrix}$

where

[MF169]

_(a,b) ^(d) g[k]

and

[MF170]

_(b,−a) ^(J,d) G[

],a,b,k,

∈

denote discrete analogues of von Neumann's TFSOs in the TD and FD in(4), respectively defined by

[MF171]

$\begin{matrix}\left. \begin{matrix}{{{_{a,b}^{d}{z\lbrack k\rbrack}} = {{z\left\lbrack {k - a} \right\rbrack}W^{- {b{({k - \frac{a}{2}})}}}}},a,b,k,{\in {\mathbb{Z}}},} \\{{{_{b,{- a}}^{1,d}{Z\lbrack \rbrack}} = {{Z\left\lbrack { - b} \right\rbrack}W^{a{({ - \frac{b}{2}})}}}},a,b,,{\in {{\mathbb{Z}}.}}}\end{matrix} \right\} & (24)\end{matrix}$

The TD-signature v[k;χ] contains N′ TD-templates of type-3

[MF172]

u _(m′) ⁽³⁾[k;X], 1≤m′≤N′,

while the FD-signature

[MF173]

V[

;χ]

does N FD-templates of type-4

[MF174]

U _(m) ⁽⁴⁾[

;X′], 1≤m≤N

so that the CCF between such a signature and its embedded template mayhave a large value via the phase coding. Note that the TD-template

[MF175]

u _(m′) ⁽³⁾[k;X]

has a rectangular support NMΔt×LΔf in the TFP, while the FD-template

[MF176]

U _(m) ⁽⁴⁾[

;X′]

has a support LΔt×N′M′Δf. Substituting the compositions of TFSOs

[MF177]

_(0,m′M) ^(d),

_(m,M,0) ^(d) g[k]

(resp.

[MF178]

_(0,−mM) ^(f,d)

_(m′M′,0) ^(f,d) G′[

]

) into (22) shows that the TD-signature and FD-signature

[MF179]

$\begin{matrix}\left. \begin{matrix}{{{v\left\lbrack {k;} \right\rbrack} = {\frac{1}{\sqrt{{NN}^{\prime}}}{\sum\limits_{m = 0}^{N - 1}{\sum\limits_{m^{\prime} = 0}^{N^{\prime} - 1}{X_{m}X_{m^{\prime}}^{\prime}_{{mM},{m^{\prime}M^{\prime}}}^{d}{g\lbrack k\rbrack}}}}}},} \\{{V\left\lbrack {;} \right\rbrack} = {\frac{1}{\sqrt{{NN}^{\prime}}}{\sum\limits_{m = 0}^{N - 1}{\sum\limits_{m^{\prime} = 0}^{N^{\prime} - 1}{X_{m}X_{m^{\prime}}^{\prime}_{m^{\prime}M^{\prime}}^{f,d}{G\lbrack \rbrack}}}}}}\end{matrix} \right\} & (25)\end{matrix}$

are perfectly symmetrical. Suppose that a radar TD-signal s[k;χ] withits CE

[MF180]

ψ[k;χ]

and the carrier

[MF181]

$_{c} = \left\lfloor \frac{\Omega}{2\; \pi \; \Delta \; f} \right\rfloor$

and its FT, FD-signal S[l;χ] have the form

[MF182]

$\begin{matrix}\left. \begin{matrix}{{{s\left\lbrack {k;} \right\rbrack} = {{\psi \left\lbrack {k;} \right\rbrack}W^{{{- _{c}}k},}}},} \\{{{S\left\lbrack {;} \right\rbrack} = {\frac{1}{2}\left( {{\Psi \left\lbrack {{ - _{c}};} \right\rbrack} + {\Psi^{*}\left\lbrack {{{- } - _{c}};} \right\rbrack}} \right)}},{{\Psi \left\lbrack {;} \right\rbrack} = {{\mathcal{F}^{d}\left\lbrack {\psi \left\lbrack {k;} \right\rbrack} \right\rbrack}.}}}\end{matrix} \right\} & (26)\end{matrix}$

Writing the CE and its DFT in the form

[MF183]

$\begin{matrix}\left. \begin{matrix}{{{\psi \left\lbrack {k;} \right\rbrack} = {\frac{1}{\sqrt{{PP}^{\prime}}}{\sum\limits_{q,{q^{\prime} = 0}}^{{P - 1},\; {P^{\prime} - 1}}{{d_{\overset{\_}{q}} \cdot _{{qNM},\; {q^{\prime}N^{\prime}M^{\prime}}}^{d}}{v\left\lbrack {k;} \right\rbrack}}}}},} \\{{{\Psi \left\lbrack {;} \right\rbrack} = {\frac{1}{\sqrt{{PP}^{\prime}}}{\sum\limits_{q,{q^{\prime} = 0}}^{{P - 1},\; {P^{\prime} - 1}}{{d_{\overset{\_}{q}} \cdot _{{q^{\prime}N^{\prime}M^{\prime}},\; {- {qNM}}}^{f,d}}{V\left\lbrack {;} \right\rbrack}}}}},}\end{matrix} \right\} & (27)\end{matrix}$

one can design the radar TD- and FD-signals

[MF184]

s[k;χ],S[

;χ].

This is a 2-dimensional train of PP′ non-overlapped signatures

[MF185]

v[k;χ]

(resp.

[MF186]

V[

;χ])

of duration T_(s)=NMΔt and carrier-spacing F_(s)=N′M′Δf, where

[MF187]

d _({right arrow over (q)})ϵ

is a data symbol on the lattice

[MF188]

T _(s)

×F _(s)

of the TFP, with address

[MF189]

{right arrow over (q)}=(q,q′).

Namely, a radar system needs PT_(s)×P′F_(s) time-duration-bandwidth tosearch for targets whose delay t_(d)∈(0, PT_(s)) and Doppler shift.

[MF190]

${f_{D} \in \left( {{- \frac{W_{D}}{2}},\frac{W_{D}}{2}} \right)},{W_{D} = {P^{\prime}F_{s}}}$

are not known in advance (simply set d_(q→)=1), while a datacommunication system sends P·P′

[MF191]

-ary data

(e.g.,

[MF192]

$\left. {{d_{\overset{\rightarrow}{q}} \in \left\{ e^{i\frac{2\pi}{\mathcal{M}}k} \right\}_{k = 0}^{\mathcal{M} - 1}},{1 \leq q \leq P},{1 \leq q^{\prime} \leq P^{\prime}}} \right).$

Suppose now that such a signal s[k;χ] is transmitted through the channelwith

[MF193]

${\theta^{\prime,d} = \left( {k_{d},_{D}} \right)},{k_{d} = \left\lfloor \frac{t_{d}}{\Delta \; t} \right\rfloor},{_{D} = {\left\lfloor \frac{f_{D}}{\Delta f} \right\rfloor.}}$

Using [MF194]

modulation

_(0,l) _(c) ^(d), demodulation

_(0,−l) _(c) ^(d),

one can obtain its received TD signal, being demodulated at a mixer or ahomodyne receiver

[MF195]

$\begin{matrix}\left. \begin{matrix}{{{r\left\lbrack {{k;},A,\kappa,\theta^{\prime,d}} \right\rbrack} = {{A\; \; e^{i\; \kappa}{\psi_{r}\lbrack k\rbrack}} + {\xi \lbrack k\rbrack}}},} \\{{\psi_{r}\lbrack k\rbrack} = {{_{0,{- _{c}}}^{d}_{k_{d},_{D}}^{d}_{0,_{c}}^{d}{\psi \left\lbrack {k;} \right\rbrack}} =}} \\{{{{{W^{k_{d}_{c}}_{k_{d},_{D}}^{d}\psi};}\left\lbrack {k;} \right\rbrack},}}\end{matrix} \right\} & (28)\end{matrix}$

where

[MF196]

ψ_(r)[k]

is the signal component CE in the received signal, η[k] interference,and ξ[k] Gaussian noise. The FD expression, its DFT

[MF197]

R[

;χ,A,κ,θ′ ^(,d)]=

^(d)[r[k;χ,A,κ,θ′ ^(,d)]].

is omitted here. Its PD

[MF198]

may be absorbed into e^(iK) by re-definition, but should be canceled-outat the correlation receiver as discussed below. Such received TD and FDsignals provide the data w and its DFT W, observed in the mixer.

Independent and identically distributed (i.i.d.) TD- and FD-PCs giveindependent N′ TD-templates

[MF199]

u _(m′) ⁽³⁾[k;X], 1≤m′≤N′

and N FD-templates [MF200]

U _(m) ⁽⁴⁾[

;X′], 1≤m≤N

in the M-ary detection. Note that PCs have two functions: to randomise asignal and to generate several PDs caused by the NCP of TFSs;Fortunately, such a PD itself provides a good indication for parameterestimation in the sense that the transmitted signal is endowed witheasy-traceability. This indicates the pros and cons of the use of PCs.In fact, the bandwidth of phase coded systems needs to be much greaterthan that of a classic radar system; The multi-carrier technique, i.e.,the FD-PC multiples this bandwidth by the number of sub-carriers and sorequires a super-wide-band signal.

<5. M-Ary Detection and Estimation of TD and FD Signals>

Let us employ the M-ary detection strategy

using the TD-CE in (27)

[MF201]

ψ[k;χ]

(resp. FD-CE

[MF202]

Ψ[

;χ])

to detect the radar signal in (26)

[MF203]

s[k;χ]

(resp.

[MF204]

S[

;χ]).

Consider a strategy whereby the receiver chooses among the NN′hypotheses H_(m,m′) associated with (17). It suffices to find thosevalues of the parameters θ′^(,d) for which the LLF (or the real part ofits associated CCF) in the TD (resp. the FD) is maximum. Consider firstthe problem of detecting the received TD-template CE

[MF205]

${Ae}^{i\; k}W^{{\hat{k}}_{d}\; _{c}}_{{\hat{k}}_{d},_{\mu}}^{d}{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack}$

with its address

[MF206]

(ρ′,{right arrow over (p)}),{right arrow over (p)}=(p,p′)

of the lattice in the TFP

[MF207]

T _(c)

×F _(c)

,

where

[MF208]

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾[k]=d _({right arrow over (p)})

_(pNM,p′N′M′) ^(d)

_(0,ρ′M′) ^(d) X′ _(ρ′) u _(ρ′) ⁽³⁾[k;X], 0≤ρ′≤N′−1,  (29)

is the TD-template CE of type 3 (cf. (22), (27)) with ρ′-th TD-template

[MF209]

X′ _(ρ′) u _(ρ′) ⁽³⁾[k;X]

[MF210]

{circumflex over (k)} _(d)

denotes an integer-valued estimate of k_(d),

[MF211]

_(μ)

an integer-valued parameter for estimating

[MF212]

_(D).

This CE is embedded in the estimated and received CE

[MF213]

Ae ^(iκ) W ^({circumflex over (k)}) ^(d) ^(l) ^(c)

_({circumflex over (k)}) _(d) _(,l) _(μ) ^(d)ψ[k;χ]

of

[MF214]

ψ[k;χ]

in (27) (cf. ψ_(r)[k] in (28)), in which the relation

[MF215]

=

is used. Equation (29) shows that the CE contains a consequential phase,caused by several PDs as given below. Such a phase should becanceled-out in a CCF. Denote the CE complementary to

[MF216]

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾[k]

by

[MF217]

ψ_(η,ρ′,{right arrow over (p)}) ⁽³⁾[k], 0≤ρ′≤N′−1,

defined as

[MF218]

ψ_(η,ρ′,{right arrow over (p)}) ⁽³⁾[k]=Σ_(m′=0,m′≠ρ′)^(N′−1)ψ_(m′,{right arrow over (p)}) ⁽³⁾[k],

ψ_(m′,{right arrow over (p)}) ⁽³⁾[k]=d _({right arrow over (p)})

_(ρNM,p′N′M′) ^(d)

_(0,m′M′) ^(d) X′ _(m′) u _(m′) ⁽³⁾[k;X], 0≤m′≤N′−1.

Equations (22) and (27) indicate that N′ TD-templates of type 3

[MF219]

u _(ρ′) ⁽³⁾[k;X], 0≤ρ′≤N′−1

are available and the receiver is to use N′ TD-template CEs of type 3

[MF220]

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾[k], 0≤ρ′≤N′−1

and decide which of the N′ TD-LLFs is largest.

i) Discrete-time signal detection and Doppler-ML estimation problem inthe TD: On the basis of measured values of the N_(T) random variablesw=(w[0], . . . , w[N_(T)−1]), the receiver must choose between twohypotheses,

[MF221]

$\begin{matrix}{\left. \begin{matrix}{H_{0}\text{:}} & {{{\psi_{w}\lbrack k\rbrack} = {\psi_{n}\lbrack k\rbrack}},} \\{H_{1}\text{:}} & {{\psi_{w}\lbrack k\rbrack} = {{Ae}^{ik}W^{{\hat{k}}_{d}_{c}}_{{\hat{k}}_{d},_{\mu}}^{d}\frac{1}{\sqrt{{PP}^{\prime}}}{\sum\limits_{{p = 0},{p^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}\; \left( {{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack} +} \right.}}} \\\; & {{\left. {\psi_{\eta,\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack} \right) + {\psi_{n}\lbrack k\rbrack}},}\end{matrix} \right\} \mspace{20mu} {{0 \leq k \leq {N_{T} - 1}},{0 \leq \rho^{\prime} \leq {N^{\prime} - 1}},}} & (30)\end{matrix}$

where

[MF222]

ψ_(w)[k]

is the NB CE of the observation data

[MF223]

${{w\lbrack k\rbrack} = {{\psi_{w}\lbrack k\rbrack}W^{{- _{r}}k}}},{_{r} = \left\lfloor \frac{\Omega_{r}}{2{\pi\Delta}\; f} \right\rfloor}$

at time k,

[MF224]

ψ_(n)[k]

the NB CE of white Gaussian noise

[MF225]

n[k]=

ψ_(D)[k]

[MF226]

N _(T) =└T/Δt┘>>1

the number of samples during the observation duration (0,T). Note thatthe signal component in hypothesis H₁ is equal to

[MF227]

(N′)^(1/2) Ae ^(iκ)

ψ[k;χ].

Suppose that N′ TD-template CEs of type 3

[MF228]

ψ_(m′,{right arrow over (p)}) ⁽³⁾[k], 0≤m′≤N′−1

of equal energies are quasi-orthogonal in the sense that

[MF229]

${{\sum\limits_{k = 0}^{N_{T} - 1}\; {{\psi_{m_{1}^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack}{\psi_{m_{2}^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack}}}1},{m_{1}^{\prime} \neq {m_{2}^{\prime}.}}$

Then one can obtain:

Proposition 4: ii) Signal detection and Doppler-ML estimation problem inthe TD:

On the basis of observed data w=(w[0], . . . , w[N_(T)−1]) in thepresence of white Gaussian noise with unilateral spectral density N₀,the logarithm of the

[MF230]

(ρ′,{right arrow over (p)})

-th TD-LF for detecting and estimating the TD-template CE of type 3

[MF231]

Ae ^(iκ)

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾[k]

is given as[21]

[MF232]

$\begin{matrix}\left. \begin{matrix}{{{\ln \mspace{14mu} {\Lambda_{\rho^{\prime},\overset{\rightarrow}{p}}\left\lbrack {{{w\lbrack k\rbrack};A},\kappa,{{\hat{k}}_{d};_{\mu}}} \right\rbrack}} = {{_{\rho^{\prime},\overset{\rightarrow}{p}}\left( {A,\kappa,{{\hat{k}}_{d};_{\mu}}} \right)} - \frac{d_{\rho^{\prime},\overset{\rightarrow}{p}}^{2}}{2}}},} \\{{_{\rho^{\prime},\overset{\rightarrow}{p}}\left( {A,\kappa,{{\hat{k}}_{d};_{\mu}}} \right)} =} \\{{\frac{{Ae}^{- {ik}}W^{{- {\hat{k}}_{d}}_{c}}}{N_{0}\sqrt{{PP}^{\prime}}}{\sum\limits_{p,{p^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}\; {\Delta \; t{\sum\limits_{k = 0}^{N_{T} - 1}\; {\left( {_{{\hat{k}}_{d},_{\mu}}^{d}{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack}} \right)^{*}W^{\alpha {({{k:{\hat{k}}_{d}},_{\mu}})}}{\psi_{w}\lbrack k\rbrack}}}}}},} \\{{d_{\rho^{\prime},\overset{\rightarrow}{p}}^{2} = {\frac{A^{2}}{N_{0}\sqrt{{PP}^{\prime}}}{{\sum_{p,{p^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}\; {\Delta \; t{\sum_{\hat{k} = 0}^{N_{T} - 1}\; {_{{\hat{k}}_{d},_{\mu}}^{d}{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack}}}}}}^{2}}},}\end{matrix} \right\} & (31)\end{matrix}$

where

[MF233]

${\alpha \left( {{k\text{:}\mspace{14mu} {\hat{k}}_{d}},_{\mu}} \right)} = {{- {_{\mu}\left( {k - \frac{{\hat{k}}_{d}}{2}} \right)}} + {{_{D}\left( {k - {\hat{k}}_{d}} \right)}.}}$

Let

[MF234]

ρ′=ρ′₀ and

_(μ)=

*_(μ)

be integers for a given decision level r₀ satisfying

[MF235]

$\begin{matrix}{\mspace{79mu} {{{\max\limits_{\rho^{\prime},_{\mu}}\frac{{_{\rho^{\prime},\overset{\rightarrow}{p}}^{\prime}\left( {\hat{A},\hat{\kappa},{{\hat{k}}_{d};_{\mu}}} \right)}}{d_{\rho^{\prime},\overset{\rightarrow}{p}}}} > r_{0}},{1 \leq \rho^{\prime} \leq N^{\prime}},{{_{\rho^{\prime},\overset{\rightarrow}{p}}^{\prime}\left( {\hat{A},\hat{\kappa},{{\hat{k}}_{d};_{\mu}}} \right)} = {\frac{{Ae}^{- {ik}}W^{{- {\hat{k}}_{d}}_{c}}}{\hat{A}\; e^{{- i}\; \hat{k}}N_{0}\sqrt{{PP}^{\prime}}}{\sum\limits_{p,{p^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}\; {\Delta \; t{\sum\limits_{k = 0}^{N_{T} - 1}\; {\left( {_{{\hat{k}}_{d},_{\mu}}^{d}{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack}} \right)^{*}W^{\alpha {({{k:{\hat{k}}_{d}},_{\mu}})}}{\psi_{w}\lbrack k\rbrack}}}}}}},}} & (32)\end{matrix}$

where

[MF236]

g′ _(ρ′,{right arrow over (p)})(Â,{circumflex over (κ)},{circumflex over(k)} _(d);

_(μ))

is a compensated variant of the statistic in (31)

[MF237]

g _(ρ′,{right arrow over (p)})(A,κ,{circumflex over (k)} _(d);

_(μ))

by the MLE in (16) [MF238]

Âe ^(−i{circumflex over (κ)})

of Ae^(−iκ). Then the receiver decides the

[MF239]

(ρ′,{right arrow over (p)})

-th CE has been arrived at the address (p′,p^(→)) of the lattice

[MF240]

T _(c)

×F _(c)

in the TFP and if all statistics

[MF241]

|g′ _(ρ′,{right arrow over (p)})(k _(d) ,A,κ;

_(μ))|/d _(ρ′,{right arrow over (p)})

lie below r₀, the receiver decides that no signal was transmitted. Thus

[MF242]

*_(μ)

is an ML estimate of

[MF243]

_(μ)

for a given

[MF244]

{circumflex over (k)} _(d),

in which the use of the operator

[MF245]

needs the phase function

[MF246]

Next one moves on the detection and delay-estimation problem in the FDusing the DFT of the measured w,

[MF247]

${{W\lbrack \rbrack} = {{\mathcal{F}^{d}\left\lbrack {w\lbrack k\rbrack} \right\rbrack} = {\frac{1}{\sqrt{N_{T}}}{\sum\limits_{k = 0}^{N_{T} - 1}\; {{w\lbrack k\rbrack}W^{kl}}}}}},{0 \leq  \leq {N_{T} - 1.}}$

Let us consider the problem of detecting the received FD-template CE

[MF248]

Ae ^(iκ)

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾

with its address

[MF249]

(ρ,{right arrow over (p)})

of the lattice

[MF250]

T _(c)

×F _(c)

,

where

[MF251]

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾[

]=d _({right arrow over (p)})

_(p′N′M′,−pNM) ^(f,d)

_(0,−ρM) ^(f,d) X _(ρ) U _(ρ) ⁽⁴⁾[

;X′], 0≤ρ≤N−1  (33)

is the FD-template CE of type 4 (cd. (22), (27)) with ρ-th FD-templateof type 4

[MF252]

X _(ρ) U _(ρ) ⁽⁴⁾[

:X′]

This CE is embedded in the estimated and received FD-CE

[MF253]

Ae ^(iκ)

Ψ[

;χ]

of the FD-CE, i.e., the DFT in (27) [MF254]

Ψ[

;χ],

where

[MF255]

_(D)

is an integer-valued estimate of

[MF256]

_(D),

k_(σ) an integer-valued parameter for estimating k_(d), and the relation

[MF257]

=

is used. Denote the CE complementary to

[MF258]

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾[

]

by

[MF259]

Ψ_(η,ρ,{right arrow over (p)}) ⁽⁴⁾[

], 0≤ρ≤N−1,

[MF260]

$\begin{matrix}\left. \begin{matrix}{{{\Psi_{\eta,\rho,\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack} = {\sum_{{m = 0},{m \neq \rho}}^{N - 1}\; {\Psi_{m,\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack}}},} \\{{{\Psi_{m,\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack} = {d_{\overset{\rightarrow}{p}}_{{p^{\prime}N^{\prime}M^{\prime}},{- {pNM}}}^{f,d}_{0,{- {mM}}}^{f,d}X_{m}{U_{m}^{(4)}\left\lbrack {;X^{\prime}} \right\rbrack}}},{0 \leq m \leq {N - 1.}}}\end{matrix} \right\} & (34)\end{matrix}$

Equations (22) and (27) suggest that N FD-templates of type 4

[MF261]

U _(ρ) ⁽⁴⁾[

;X′], 0≤ρ≤N−1

are available and the receiver is to use N FD-template CEs of type 4

[MF262]

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾[

], 0≤ρ≤N−1

and decide which of the N LLFs is largest (FIG. 3b ).

ii) Signal detection and delay-ML estimation problem in the FD:

On the basis of observed W=(W[0], . . . , W[N_(T)−1]), the receiver mustchoose between two hypotheses in the FD,

[MF263]

$\begin{matrix}{\left. \begin{matrix}{H_{0}^{\prime}\text{:}} & {{{W\lbrack \rbrack} = {N\lbrack \rbrack}},} \\{H_{1}^{\prime}\text{:}} & {{{W\lbrack \rbrack} = {{\frac{{Ae}^{ik}W^{k_{\sigma}_{c}}}{\sqrt{{PP}^{\prime}}}_{{\hat{}}_{D},k_{\sigma}}^{f,d}{\sum\limits_{{p = 0},{p^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}\; \left( {{S_{\rho,\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack} + {E_{\rho,\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack}} \right)}} + {N\lbrack \rbrack}}},}\end{matrix} \right\} \mspace{20mu} {{0 \leq  \leq {N_{B} - 1}},{0 \leq \rho \leq {N - 1}},}} & (35)\end{matrix}$

where

[MF264]

W[

]=

^(d)[

ψ_(w)[k]

]

is the DFT of the observed data w[k] with the NB CE

[MF265]

ψ_(w)[k].

[MF266]

N[

]=

^(d)[

ψ_(D)[k]

]

the DFT of the noise n[k] with the NB CE

[MF267]

ψ_(n)[k].

[MF268]

N _(B) =└B/Δf┘

the sample number of the bandwidth B, for simplicity N_(B)=N_(T);

[MF269]

S _(ρ,{right arrow over (p)}) ⁽⁴⁾[

]=

^(d)[

^(−1,d)[Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾[

]

]]

the

[MF270]

(ρ,{right arrow over (p)})

-th template signal spectrum with its FD-template CE of type 4 in (33)

[MF271]

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾[

],

and

[MF272]

E _(ρ,{right arrow over (p)}) ⁽⁴⁾[

]=

^(d)[

^(−1,d)[Ψ_(η,ρ,{right arrow over (p)}) ⁽⁴⁾[

]

]]

its complement spectrum with CE in (34)

[MF273]

Ψ_(η,ρ,{right arrow over (p)}) ⁽⁴⁾[

].

Note that the signal component in hypothesis H′₁ is equal to

[MF274]

(N)^(1/2) Ae ^(iκ)

Ψ[

;χ].

Suppose that N FD-template CEs of type 4 (cf. (22), (27))

[MF275]

Ψ_(m,{right arrow over (p)}) ⁽⁴⁾[

], 0≤m≤N−1

of equal energies are quasi-orthogonal in the sense that

[MF276]

${{\sum\limits_{ = 0}^{N_{B} - 1}\; {{\Psi_{m_{1},\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack}{\Psi_{m_{2},\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack}}}1},{m_{1} \neq {m_{2}.}}$

Then one can obtain:

Proposition 5: On the basis of data

[MF277]

W=(W[0], . . . , W[N _(T)−1]), W[

]=

^(d)[w[k]]

in the presence of white Gaussian noise with unilateral spectral densityN₀, the logarithm of the

[MF278]

(ρ,{right arrow over (p)})

-th FD-LF for detecting and estimating

[MF279]

S _(ρ,{right arrow over (p)}) ⁽⁴⁾[

]

is given as

[MF280]

$\begin{matrix}{\mspace{79mu} {{{\ln \mspace{14mu} {\Lambda_{\rho,\overset{\rightarrow}{p}}^{\prime}\left\lbrack {{{W\lbrack \rbrack};A},\kappa,{{\hat{}}_{D};k_{\sigma}}} \right\rbrack}} = {{G_{\rho,\overset{\rightarrow}{p}}\left( {A,\kappa,{{\hat{}}_{D};k_{\sigma}}} \right)} - \frac{D_{\rho,\overset{\rightarrow}{p}}^{2}}{2}}},\mspace{20mu} {D_{\rho,\overset{\rightarrow}{p}}^{2} = {\frac{A^{2}}{N_{0}\sqrt{{PP}^{\prime}}}{{\sum\limits_{p,{p^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}\; {\Delta \; f{\sum\limits_{ = 0}^{N_{B} - 1}\; {_{{\hat{}}_{D},{- k_{\sigma}}}^{f,d}{S_{\rho,\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack}}}}}}^{2}}},{{G_{\rho,\overset{\rightarrow}{p}}\left( {A,\kappa,{{\hat{}}_{D};k_{\sigma}}} \right)} = {\frac{{Ae}^{- {ik}}W^{{- k_{\sigma}}_{c}}}{N_{0}\sqrt{{PP}^{\prime}}}{\sum\limits_{p,{p^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}\; {\times \Delta \; f{\sum\limits_{ = 0}^{N_{B} - 1}\; {\left( {_{{\hat{}}_{D},{- k_{\sigma}}}^{f,d}{S_{\rho,\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack}} \right)^{*}W^{\beta {({{:{\hat{}}_{D}},k_{\sigma}})}}{W\lbrack \rbrack}}}}}}},}} & (36)\end{matrix}$

where

[MF281]

${\beta \left( {{\text{:}\mspace{14mu} {\hat{}}_{D}},k_{\sigma}} \right)} = {{k_{\sigma}\left( { - \frac{{\hat{}}_{D}}{2}} \right)} - {{k_{d}\left( { - {\hat{}}_{D}} \right)}.}}$

Let [MF282]

ρ=ρ₀ and k _(σ) =k* _(σ)

be integers satisfying

[MF283]

$\begin{matrix}{\mspace{79mu} {{{\max\limits_{\rho,k_{\sigma}}\frac{{G_{\rho,\overset{\rightarrow}{p}}^{\prime}\left( {A,\kappa,{{\hat{}}_{D};k_{\sigma}}} \right)}}{D_{\rho,\overset{\rightarrow}{p}}}} > r_{0}^{\prime}},{1 \leq \rho \leq N},{{G_{\rho,\overset{\rightarrow}{p}}^{\prime}\left( {A,\hat{\kappa},{{\hat{}}_{D};k_{\sigma}}} \right)} = {\frac{{Ae}^{- {ik}}W^{{- k_{\sigma}}_{c}}}{\hat{A}\; e^{{- i}\; \hat{k}}N_{0}\sqrt{{PP}^{\prime}}}\Delta \; f{\sum\limits_{ = 0}^{N_{B} - 1}\; {\left( {_{{\hat{}}_{D},{- k_{\sigma}}}^{f,d}{S_{\rho,\overset{\rightarrow}{p}}^{(4)}\lbrack \rbrack}} \right)^{*}W^{\beta {({{:{\hat{}}_{D}},k_{\sigma}})}}{W\lbrack \rbrack}}}}}}} & (37)\end{matrix}$

for a given decision level r′₀. Then the receiver decides the(ρ,p^(→))-th signal has been arrived at the address

[MF284]

(ρ,{right arrow over (p)})

of the lattice in the TFP

[MF285]

T _(c)

×F _(c)

and if all statistics

[MF286]

|G′ _(ρ,{right arrow over (p)})(k _(σ);

_(D) ,Â,{circumflex over (κ)})|/D _(ρ,{right arrow over (p)})

lie below r′₀, the receiver decides that no signal was transmitted. Thus

[MF287]

k* _(σ)

is a ML estimate of k_(d) for a given

[MF288]

_(D),

in which the use of the operator

[MF289]

_(∈k) _(σ)

needs the phase function

[MF290]

,

an FD analogue of the

[MF291]

,

so one can get (36) and (37).

<6. TD and FD Cross-Correlations for Parameter Estimation> <6.1 TD andFD Cross-Correlations>

Suppose that the observed input

[MF292]

w(t)=

ψ_(w)(t)e ^(iΩt)

is NB and both the real and imaginary parts of the CE

[MF293]

ψ_(w)(t)

can be measured separately[18, 3].

In order to cancel out the phase factors of

[MF294]

Ae ^(iκ)

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾[k])

(resp.

[MF295]

Ae ^(w)

Ψ_(ρ′,{right arrow over (p)}) ⁽⁴⁾[

])

and get a time and frequency symmetrical statistic,

instead of the statistic in (31)

[MF296]

g _(ρ′,{right arrow over (q)})({circumflex over (k)} _(d),

_(μ))

having

[MF297]

(resp. the statistic in (36)

[MF298]

G _(ρ,{right arrow over (p)})(k _(σ),

_(D))

having

[MF299]

,

one can use its associated CCF to be defined below (see FIG. 3b ):

Lemma 1: Suppose that in two hypotheses H₀ and H₁ in (30) the CE

[MF300]

ψ_(n)[k]

is Gaussian. Then

[MF301]

<ψ₀,ψ_(ρ′,{right arrow over (p)}) ⁽³⁾>_(d,k)<<1,

where <⋅,⋅>_(d,k) denotes the IP in the space

[MF302]

₂(

)

of discrete-time TD functions. Thus one can define a CCF, called atype-3 correlator, between the received CE

[MF303]

ψ[k;χ]

(namely, the signal component of the received CE in (28)

[MF304]

ψ_(r)[k;χ]

with factor

[MF305]

Ae ^(iκ)

),

instead of the input CE

[MF306]

ψ_(w)[k;χ]

to the receiver, and the complex impulse response of a NB filter matchedto the estimated template CE with address

[MF307]

(ρ′,{right arrow over (p)})

[MF308]

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾[k]|_(d) _({right arrow over (p)}) ₌₁

given as

[MF309]

$\begin{matrix}{{c_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\left( {_{\mu};{\hat{k}}_{d}} \right)} = {A\; e^{i\; \kappa}W^{k_{a}_{c}} \times {\sum\limits_{k \in {\mathbb{Z}}}{_{k_{d},_{D}}^{d}{\psi \left\lbrack {k,\chi} \right\rbrack}\left( {W^{{\hat{k}}_{a}_{c}}_{{\hat{k}}_{d},_{\mu}}^{d}_{{pNM},{p^{\prime}N^{\prime}M^{\prime}}}^{d}_{0,{\rho^{\prime}M^{\prime}}}^{d}Y_{\rho^{\prime}}^{\prime}{u_{\rho^{\prime}}^{(3)}\left\lbrack {k:Y} \right\rbrack}} \right)^{*}}}}} & (38)\end{matrix}$

in which the TD-template CE of type 3 in (29)

[MF310]

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾[k]

and the codes X′_(ρ′),X are replaced by Y′_(ρ′),Y.

One can find after some labor that this CCF of type 3 is given as

[MF311]

$\begin{matrix}{{c_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\left( {_{\mu};{\hat{k}}_{d}} \right)} = {\frac{{Ae}^{i\; \kappa}W^{{({k_{d} - {\hat{k}}_{d}})}_{c}}}{\sqrt{{PP}^{\prime}}}{\sum\limits_{\overset{\rightarrow}{q}}{d_{\overset{\rightarrow}{q}}\frac{Y_{\rho^{l}}^{\prime,*}}{N\sqrt{{N\;}^{\prime}}}{\sum\limits_{{m = 0},{m^{\prime} = 0},{n = 0}}^{{N - 1},{N^{\prime} - 1},{N - 1}}\; {X_{m}X_{m}^{\prime}Y_{n}^{*} \times {\quad{{\theta_{gg}\left\lbrack {{\left( {{\hat{k}}_{d} - k_{d}} \right) + {\left( {p - q} \right){MN}} + {\left( {n - m} \right)M}},{\left( {_{\mu} - _{D}} \right) + {\left( {p^{\prime} - q^{\prime}} \right)M^{\prime}N^{\prime}} + {\left( {\rho^{\prime} - m^{\prime}} \right)M^{\prime}}}} \right\rbrack} \times W^{\frac{1}{2}{({{k_{d}_{\mu}} - {{\hat{k}}_{d}_{D}} + {2{MNq}\; {\upsilon_{0}{\lbrack _{\mu}\rbrack}}} - {2M^{\prime}N^{\prime}q^{\prime}{\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}}}})}} \times W^{\frac{1}{2}{({{{({{m\; \rho^{\prime}} - {m^{\prime}n}})}{MM}^{\prime}} - {{({m^{\prime} + \rho^{\prime}})}M^{\prime}{\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}}} + {{({m + n})}M\; {\upsilon_{0}{\lbrack _{\mu}\rbrack}}}})}}}}}}}}}} & (39)\end{matrix}$

Unfortunately, the AFs θ_(gg)(τ,υ) and Θ_(GG)(υ,−τ) have many sidelobesin general. A Gaussian chip-pulse g(t), however, gives a radicalsolution to the estimation problems because it has its separable andexponentially decayed AF in terms of of τ and υ

[MF312]

${{\theta_{gg}\left( {\tau,v} \right)} = {{\Theta_{GG}\left( {v,{- \tau}} \right)} = {{\exp \left( {- \frac{r^{2}}{2s_{t}^{2}}} \right)} \cdot {\exp \left( {- \frac{v^{2}}{2s_{f}^{2}}} \right)}}}},{{{with}\mspace{14mu} s_{t}s_{f}} = \frac{1}{2\pi}},$

where

[MF313]

${s_{t}^{2} = {\sum\limits_{k}{k^{2}{g\lbrack k\rbrack}}}},{s_{f}^{2} = {\sum\limits_{\xi}{^{2}{{G\lbrack \rbrack}.}}}}$

For N,N′>>1, both the first and second arguments of θ_(gg)[⋅,⋅] shouldbe relatively small, i.e., q^(→)=p^(→) so that

[MF314]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d))

has a large value; All of the terms with

[MF315]

{right arrow over (q)}≠{right arrow over (p)}

in (39) can be negligible. This property of Gaussian plays a centralrole in determining

_(μ) for maximizing

[MF316]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d)).

Evaluating threefold summation of PCs with the cisoidal factor leads usto define a summation of IDFT-type

[MF317]

${{\mathcal{F}_{m^{\prime}N^{\prime}}^{{- 1},d}\left\lbrack {X\left\lbrack m^{\prime} \right\rbrack} \right\rbrack}_{m^{\prime}}(a)_{m^{\prime}}} = {\frac{1}{\sqrt{N^{\prime}}}{\sum\limits_{m^{\prime} = 0}^{N^{\prime} - 1}{{X\left\lbrack m^{\prime} \right\rbrack}{e^{i\; 2\; \pi \frac{a\mspace{11mu} m^{\prime}}{L}}.}}}}$

symbolically denoted by a pair of the square bracket

[MF318]

_(m′,N′) ^(−1,d)[⋅]_(m′)

and the round bracket (a)_(m′) with its convenient notation(a)_(m′)=W^(−am′), and that of DFT-type

[MF319]

${{{\mathcal{F}_{m,N}^{d}\left\lbrack {x\left\lbrack m^{\prime} \right\rbrack} \right\rbrack}_{m}\left( b^{\prime} \right)_{m}} = {\frac{1}{\sqrt{N}}{\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}e^{{- i}\; 2\; \pi \frac{{mb}^{\prime}}{L}}}}}},$

symbolically denoted by a pair of convenient notations

[MF320]

_(m,N) ^(d)[⋅]_(m′)(b′)_(m) =W ^(mb′).

Using the notations:

[MF321]

ν₀[

_(μ)]=

_(μ)−

_(D), τ₀[{circumflex over (k)} _(d)]={circumflex over (k)} _(d) −k _(d),

one can obtain

Lemma 2: If the receiver of type-3 has its address

[MF322]

{right arrow over (p)}={right arrow over (q)}

and Y=X, Y′=X′, then its CCF becomes

[MF323]

$\begin{matrix}{{c_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\left\{ {_{\mu};{\hat{k}}_{d}} \right)} \simeq {\frac{{Ae}^{i\; \kappa}W^{{- {r_{0}{\lbrack{\overset{\sim}{k}}_{a}\rbrack}}}_{n}}}{\sqrt{{PP}^{\prime}}}d_{\overset{\rightarrow}{p}}X_{\rho^{\prime}}^{\prime,*}{W\left( {{{MNpv}_{0}\left\lbrack _{\mu} \right\rbrack} - {\left( {{N^{\prime}p^{\prime}} + \frac{\rho^{\prime}}{2}} \right)M^{\prime}{\tau_{0}\left\lbrack {\hat{k}}_{d} \right\rbrack}} + \frac{{k_{d}_{\mu}} - {{\hat{k}}_{d}_{D}}}{2}} \right)} \times {\quad{\mathcal{F}_{m^{\prime},N^{\prime}}^{{- 1},d}\left\lbrack {{X_{m}^{\prime}{\exp\left( {- \frac{\left( {{v_{0}\left\lbrack _{\mu} \right\rbrack} + {\left( {\rho^{\prime} - m^{\prime}} \right)M^{\prime}}} \right)^{2}}{2s_{f}^{2}}} \right)}\left. \quad{{\mathcal{F}_{m,N}^{d}\left\lbrack {X_{m}{\mathcal{F}_{n,N}^{d}\begin{bmatrix}{X_{n}^{*} \times \exp} \\\left( {- \frac{\left( {{\tau_{0}\left\lbrack {\hat{k}}_{d} \right\rbrack} + {\left( {n - m} \right)M}} \right)^{2}}{2s_{t}^{2}}} \right)\end{bmatrix}}_{n} \times \left( {x_{3}\left( {- m^{\prime}} \right)} \right)_{n}} \right\rbrack}_{m}\left( {x_{3}\left( \rho^{\prime} \right)} \right)m} \right\rbrack_{m^{\prime}}\left( \frac{M^{\prime}{r_{0}\left\lbrack {\hat{k}}_{d} \right\rbrack}}{2} \right)_{m^{\prime}}},{\quad\mspace{79mu} {{x_{3}\left( \rho^{\prime} \right)} = \frac{M\left( {{v_{0}\left\lbrack _{\mu} \right\rbrack} + {\rho^{\prime}M^{\prime}}} \right)}{2}}}} \right.}}}} & (41)\end{matrix}$

Thus one can know necessary conditions for

[MF324]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d))

to have a large value are

[MF325]

|ν₀[

_(μ)]|<3s _(f) and |τ₀[{circumflex over (k)} _(d)]|<3s _(t).

Proof. Using the round bracket symbols in summations of IDFT-type andDFT-type, one can rearrange 5 components among 6 components in thesecond twiddle factor in (39)

[MF326]

$\begin{matrix}\left. \begin{matrix}\begin{matrix}{{{W^{\frac{1}{2}{{nMv}_{0}{\lbrack _{\mu}\rbrack}}} \cdot W^{{- \frac{1}{2}}m^{\prime}{nMM}^{\prime}}} = {W^{\frac{1}{2}{{nM}{({{v_{0}{\lbrack _{\rho}\rbrack}} - {m^{\prime}M^{\prime}}})}}} = \left( {x_{3}\left( {- m^{\prime}} \right)} \right)_{n}}},} \\{{{W^{\frac{1}{2}{{mMv}_{0}{\lbrack _{\mu}\rbrack}}} \cdot W^{\frac{1}{2}m\; \rho^{\prime}{MM}^{\prime}}} = {W^{\frac{1}{2}{{mM}{({{v_{0}{\lbrack _{\mu}\rbrack}} + {\rho^{\prime}M^{\prime}}})}}} = \left( {x_{3}\left( \rho^{\prime} \right)} \right)_{m}}},}\end{matrix} \\{{W^{{- \frac{1}{2}}m^{\prime}M^{\prime}{r_{0}{\lbrack{\hat{k}}_{a}\rbrack}}} = \left( \frac{M^{\prime}{r_{0}\left\lbrack {\hat{k}}_{d} \right\rbrack}}{2} \right)_{m^{\prime}}},}\end{matrix} \right\} & (42)\end{matrix}$

and move its residue term W^(−1/2 ρ′M′τ0[k{circumflex over ( )}d]) ofthe 6 components to thefirst twiddle factor in (39). The use of the separability of the AF ofthe Gaussian proves (41).

Lemma 3: Suppose that in two hypotheses H′₀ and H′₁ in (35)

[MF327]

N[

]

is Gaussian. Then the CCF

[MF328]

<N,S _(ρ,{right arrow over (p)}) ⁽⁴⁾

<<1,

where the angular brackets <⋅,⋅

denotes the IP in the space

[MF329]

₂(

)

of discrete-frequency FD functions.

Then one can define a CCF, called a type-4 correlator, between thereceived FD-CE

[MF330]

Ae ^(iκ)

Ψ[

;χ],

i.e., the DFT of the signal component of the received CE in (28)

[MF331]

ψ_(r)[k]

with factor

[MF332]

Ae ^(iκ)

,

instead of the input CE

[MF333]

Ψ_(W)[

;χ]

to the receiver, and the FD complex impulse response of a NB filtermatched to the FD estimated template CE in (33) with address

[MF334]

(ρ,{right arrow over (p)})

[MF335]

W ^(k) ^(σ) ^(l) ^(c)

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾[

]|_(d) _({right arrow over (p)}=1) _(,X′→Y′,X) _(ρ) _(→Y) _(ρ) ,

given as

[MF336]

$\begin{matrix}{{C_{\rho,\overset{\rightarrow}{p}}^{(4)}\left( {k_{\sigma};{\hat{}}_{D}} \right)} = {A\; e^{i\; \kappa}W^{k_{d}_{c}} \times {\sum\limits_{ \in {\mathbb{Z}}}{_{,{D:{- k_{3}}}}^{f,d}{\Psi \left\lbrack {;\chi} \right\rbrack}\left( {W^{k_{\sigma}_{c}}_{{\hat{}}_{D},{\ldots \; k_{\sigma}}}^{,d}_{p^{\prime}N^{\prime}{M^{\prime}:{\ldots \mspace{11mu} {pNM}}}}^{f,d}_{0,{\ldots \mspace{11mu} \rho \; M}}^{f,d}Y_{\rho}{U_{\rho}^{(4)}\left\lbrack {:Y^{\prime}} \right\rbrack}} \right)^{*}}}}} & (43)\end{matrix}$

This CCF has the form

[MF337]

$\begin{matrix}{{C_{\rho,\overset{\rightarrow}{p}}^{(4)}\left( {k_{\sigma}:{\hat{}}_{D}} \right)} = {\frac{{Ae}^{i\; \kappa}W^{{- {r_{0}{\lbrack k_{a}\rbrack}}}_{c}}}{\sqrt{{PP}^{\prime}}}{\sum\limits_{q}{d_{\overset{\rightarrow}{q}}\frac{Y_{\rho}^{*}}{N^{\prime}\sqrt{N}}{\sum\limits_{{m = 0},{m^{\prime} = 0},{n^{\prime} = 0}}^{{N - 1},{N^{\prime} - 1},{N^{\prime} - 1}}{X_{m}X_{m^{\prime}}^{\prime}Y_{n^{\prime}}^{\prime,*} \times {\Theta_{GG}\left\lbrack {{\left( {{\hat{}}_{D} - _{D}} \right) + {\left( {p^{\prime} - q^{\prime}} \right)M^{\prime}N^{\prime}} + {\left( {n^{\prime} - m^{\prime}} \right)M^{\prime}}},{{- \left( {k_{\sigma} - k_{d}} \right)} - {\left( {p - q} \right){{MN}\left( {\rho - m} \right)}M}}} \right\rbrack} \times W^{\frac{1}{2}{({{k_{d}{\hat{}}_{D}} - {k_{\sigma}_{D}} + {2{{MNqv}_{0}{\lbrack{\hat{}}_{D}\rbrack}}} - {2M^{\prime}N^{\prime}q^{\prime}{\tau_{0}{\lbrack k_{\sigma}\rbrack}}}})}} \times W^{\frac{1}{2}{({{{({{mn}^{\prime} - {m^{\prime}\rho}})}{MM}^{\prime}} - {{({m^{\prime} + n^{\prime}})}M^{\prime}{\tau_{0}{\lbrack k_{\sigma}\rbrack}}} + {{({m + \rho})}{{Mv}_{0}{\lbrack{\hat{}}_{D}\rbrack}}}})}}}}}}}} & (44)\end{matrix}$

Similarly, for N,N′>>1, one can set [MF338]

{right arrow over (q)}={right arrow over (p)}

so as to make

[MF339]

C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k _(σ);

_(D))

large; All of the terms in (44) with

[MF340]

{right arrow over (q)}≠{right arrow over (p)}

can be negligible. Then one can get the sum of the three exponents inthe twiddle factor as above. Evaluating the threefold summation of PCstogether with the cisoidal factor, one can have

Lemma 4: If the receiver of type-4 has its address

[MF341]

{right arrow over (p)}={right arrow over (q)}

and Y=X, Y′=X′, then its CCF becomes

[MF342]

$\begin{matrix}{{{C_{\rho,\overset{\rightarrow}{p}}^{(4)}\left( {k_{\sigma}:{\hat{}}_{D}} \right)} \simeq {\frac{{Ae}^{i\; \kappa}W^{{- {r_{0}{\lbrack k_{a}\rbrack}}}_{c}}}{\sqrt{{PP}^{\prime}}}d_{\overset{\rightarrow}{p}}X_{p}^{*}W^{(\begin{matrix}\begin{matrix}{{{({{N\; p} + \frac{\rho}{2}})}{{Mv}_{0}{\lbrack{\hat{}}_{D}\rbrack}}} -} \\{{M^{\prime}N^{\prime}p^{\prime}{\tau_{0}{\lbrack k_{\sigma}\rbrack}}} +}\end{matrix} \\\frac{{k_{d}{\hat{}}_{D}} - {k_{\sigma}_{D}}}{2}\end{matrix})} \times {\mathcal{F}_{m,N}^{d}\left\lbrack {X_{m}{\exp\left( {- \frac{\begin{pmatrix}{{\tau_{0}\left\lbrack k_{d} \right\rbrack} +} \\{\left( {\rho - m} \right)M}\end{pmatrix}^{2}}{2s_{t}^{2}}} \right)}} \right\rbrack}{\mathcal{F}_{m^{\prime},N^{\prime}}^{{- 1},d}\left\lbrack {X_{m^{\prime}}^{\prime}{\mathcal{F}_{n,N}^{{- 1},d}\left\lbrack {X_{n}^{\prime,*} \times {\exp\left( {- \frac{\begin{pmatrix}{{v_{0}\left\lbrack {\hat{}}_{D} \right\rbrack} +} \\{\left( {n^{\prime} - m^{\prime}} \right)M^{\prime}}\end{pmatrix}^{2}}{2s_{f}^{2}}} \right)}} \right\rbrack}_{n^{\prime}} \times \left( {x_{4}\left( {- m} \right)} \right)_{n^{\prime}}} \right\rbrack}_{m}\left( {x_{4}(\rho)} \right)_{m^{\prime}}\left( \frac{{Mv}_{0}\left\lbrack _{D} \right\rbrack}{2} \right)_{m}}},\mspace{20mu} {{x_{4}(\rho)} = {\frac{M^{\prime}\left( {{\tau_{0}\left\lbrack k_{\sigma} \right\rbrack} + {\rho \; M}} \right)}{2}.}}} & (45)\end{matrix}$

Thus we know necessary conditions for

[MF343]

C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k _(σ);

_(D))

to have a large value are

[MF344]

|ν₀[

_(D)]|<3S _(f) and |τ₀[k _(σ)]|<3s _(t).

Proof. Similarly using the round bracket symbols in summations ofIDFT-type and DFT-type, one can rearrange 5 components among 6components in the second twiddle factor in (44)

[MF345]

$\begin{matrix}\left. \begin{matrix}\begin{matrix}{{{W^{{- \frac{1}{2}}n^{\prime}M^{\prime}{r_{0}{\lbrack k_{\sigma}\rbrack}}} \cdot W^{\frac{1}{2}n^{\prime}{mMM}^{\prime}}} = {W^{{- \frac{1}{2}}n^{\prime}{M^{\prime}{({{r_{0}{\lbrack k_{\sigma}\rbrack}} - {mM}})}}} = \left( {x_{4}\left( {- m} \right)} \right)_{n^{\prime}}}},} \\{{{W^{{- \frac{1}{2}}m^{\prime}M^{\prime}{r_{0}{\lbrack k_{\sigma}\rbrack}}} \cdot W^{{- \frac{1}{2}}{m\;}^{\prime}\rho \; {MM}^{\prime}}} = {W^{{- \frac{1}{2}}m^{\prime}{M^{\prime}{({{r_{0}{\lbrack k_{\sigma}\rbrack}} + {\rho \; M}})}}} = \left( {x_{4}(\rho)} \right)_{m^{\prime}}}},}\end{matrix} \\{{W^{\frac{1}{2}{{mMv}_{0}{\lbrack{\hat{}}_{D}\rbrack}}} = \left( \frac{{Mv}_{0}\left\lbrack {\hat{}}_{D} \right\rbrack}{2} \right)_{m}},}\end{matrix} \right\} & (46)\end{matrix}$

and move its residue term W^(1/2 ρMν0[l{circumflex over ( )}D]) of the 6components to the first twiddle factor in (44). The use of theseparability of the AF of the Gaussian proves (45).Equations (41) and (45) show that correlator of type 3

[MF346]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d))

and the one of type 4

[MF347]

C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k _(σ):

_(D))

are perfectly symmetrical in terms of

[MF348]

d _(d) and

_(D).

Interchanging the TD-PC and FD-PC in the pair of type 3

[MF349]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d))

and of type 4

[MF350]

C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k _(σ);

_(D)),

called a complementary pair (CP)[27] gives other pair of type 1

[MF351]

c _(ρ,{right arrow over (p)}) ⁽¹⁾(

_(μ) ;{circumflex over (k)} _(d))

and of type 2

[MF352]

C _(ρ′,{right arrow over (p)}) ⁽²⁾(k _(σ);

_(D)),

called an original pair (OP)[24, 30], defined as follows.

[Original pair (OP) of TD and FD correlators]

Observe another decomposition of v[k;χ] and

[MF353]

C[

;χ]

given as

[MF354]

$\begin{matrix}\left. \begin{matrix}{{{v\left\lbrack {k;\chi} \right\rbrack} = {\frac{1}{\sqrt{N}}{\sum\limits_{m = 0}^{N - 1}{X_{m}_{{mM},0}^{d}{u_{m}^{(1)}\left\lbrack {k;X^{\prime}} \right\rbrack}}}}},} \\{{{V\left\lbrack {;\chi} \right\rbrack} = {\frac{1}{\sqrt{N^{\prime}}}{\sum\limits_{m^{\prime} = 0}^{N^{\prime} - 1}{X_{m}^{\prime}_{{m^{\prime}M^{\prime}},0}^{f,d}{U_{m^{\prime}}^{(2)}\left\lbrack {;X} \right\rbrack}}}}},}\end{matrix} \right\} & (47)\end{matrix}$

whose TD-template of type-1

[MF355]

u _(m) ⁽¹⁾[k;X′]

and FD-template of type-2

[MF356]

U _(m′) ⁽²⁾[

;X]

are respectively defined by

[MF357]

$\begin{matrix}\left. \begin{matrix}{{{u_{m}^{(1)}\left\lbrack {k;X^{\prime}} \right\rbrack} = {\frac{1}{\sqrt{N^{\prime}}}{\sum\limits_{m^{\prime} = 0}^{N^{\prime} - 1}{X_{m^{\prime}}^{\prime}e^{i\; \pi \; m\; m^{\prime}{MM}^{\prime}\Delta \; t\; \Delta \; f}_{0,{m^{\prime}M^{\prime}}}^{d}}}}},{g\lbrack k\rbrack},{0 \leq m \leq {N - 1}},} \\{{{U_{m}^{(2)}\left\lbrack {;X} \right\rbrack} = {\frac{1}{\sqrt{N}}{\sum\limits_{m = 0}^{N - 1}{X_{m^{\prime}}e^{{- i}\; \pi \; m\; m^{\prime}{MM}^{\prime}\Delta \; t\; \Delta \; f}_{0,{- {mM}}}^{f,d}}}}},{G\lbrack \rbrack},{0 \leq m^{\prime} \leq {N^{\prime} - 1.}}}\end{matrix} \right\} & (48)\end{matrix}$

First let [MF358]

ψ_(ρ,{right arrow over (p)}) ⁽¹⁾[k]

be the (ρ,p^(→))-th estimated and received CE having the TD-template

[MF359]

X _(ρ) u _(ρ) ⁽¹⁾[k,X′]

with address

[MF360]

(ρ,{right arrow over (p)})

of the lattice

[MF361]

T _(c)

×F _(c)

defined by the TD-template CE of type 1

[MF362]

ψ_(ρ,{right arrow over (p)}) ⁽¹⁾[k]=d _({right arrow over (p)})

_(ρNM,p′N′M′) ^(d)

_(ρM,N) ^(d) X _(ρ) u _(ρ) ⁽¹⁾[k,X′].  (49)

Then one can get its associated CCF, called a type-1 correlator

[MF363]

$\begin{matrix}{{{c_{\rho,\overset{\rightarrow}{p}}^{(1)}\left( {_{\mu}:{\hat{k}}_{d}} \right)} = {{\langle{{A\; e^{i\; \kappa}W^{k_{d}_{c}}_{k_{d},_{D}}^{d}{\psi \left\lbrack {k;\chi} \right\rbrack}},\left. {W^{{\hat{k}}_{d}_{c}} \times _{{\hat{k}}_{d},_{\mu}}^{d}{\psi_{\rho,\overset{\rightarrow}{p}}^{(1)}\left\lbrack {{k:{\hat{k}}_{d}},_{\mu}} \right\rbrack}} \right|_{{d_{p} = 1},{X^{\prime}\rightarrow Y^{\prime}},{X_{\rho}\rightarrow Y_{\rho}}}}\rangle}d}},k} & (50)\end{matrix}$

and one can have

[MF364]

$\begin{matrix}{{c_{\rho,\overset{\rightarrow}{p}}^{(1)}\left( {_{\mu}\text{:}{\hat{k}}_{d}} \right)} = {{{Ae}^{i\; \kappa}W^{k_{d}_{c}}{\sum\limits_{k \in {\mathbb{Z}}}{_{k_{d},_{D}}^{d}{\psi \left\lbrack {\hat{k};\chi} \right\rbrack}{Y_{\rho}^{*}\left( {W^{{\hat{k}}_{d}_{c}}_{{\hat{k}}_{d},_{\mu}}^{d}_{{pNM},{p^{\prime}N^{\prime}M^{\prime}}}^{d}_{{\rho \; M},0}^{d}{u_{\rho}^{(1)}\left\lbrack {k\text{:}Y^{\prime}} \right\rbrack}} \right)}^{*}}}} = {{Ae}^{j\; \kappa}W^{{({k_{d} - {\hat{k}}_{d}})}_{c}}{\sum\limits_{ij}{d_{q}\frac{Y_{\rho}^{*}}{N^{\prime}\sqrt{N}}{\sum\limits_{m,m^{\prime},n^{\prime}}^{{N - 1},{N^{\prime} - 1},{N^{\prime} - 1}}{X_{m}X_{m^{\prime}}^{\prime}Y_{n^{\prime}}^{\prime,*} \times W^{\frac{1}{2}{({{k_{d}_{\mu}} - {{\hat{k}}_{d}_{D}} + {2{{MNpv}_{0}{\lbrack _{\mu}\rbrack}}} - {2M^{\prime}N^{\prime}p^{\prime}{\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}}}})}} \times W^{\frac{1}{2}{({{{({{mn}^{\prime} - {m^{\prime}\; \rho}})}{MM}^{\prime}} - {{({m^{\prime} + n^{\prime}})}M^{\prime}{\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}}} + {{({m + \rho})}{{Mv}_{0}{\lbrack _{\mu}\rbrack}}}})}} \times {{\theta_{zz}\left\lbrack {{{\left( {p - q} \right){MN}} + {\hat{k}}_{d} - k_{d} + {\left( {\rho - m} \right)M}},{{\left( {p^{\prime} - q^{\prime}} \right)M^{\prime}N^{\prime}} + _{\mu} - _{D} + {\left( {n^{\prime} - m^{\prime}} \right)M^{\prime}}}} \right\rbrack}.}}}}}}}} & (51)\end{matrix}$

Let Y=X, Y′=X′ and discard all the terms

[MF365]

d _({right arrow over (q)})

except the term satisfying

[MF366]

{right arrow over (q)}={right arrow over (p)}.

Then one can have

[MF367]

$\begin{matrix}{{{{c_{\rho,\overset{\rightarrow}{p}}^{(1)}\left( {_{\mu}\text{:}{\hat{k}}_{d}} \right)}\text{|}} \simeq {\frac{{Ae}^{i\; \kappa}W^{{- {r_{0}{\lbrack{\hat{k}}_{d}\rbrack}}}_{c}}d_{\overset{\rightarrow}{p}}}{\sqrt{{PP}^{\prime}}}X_{\rho}^{*}{W\left( {{\left( {{N\; p} + \frac{\rho}{2}} \right){{Mv}_{0}\left\lbrack _{\mu} \right\rbrack}} - {M^{\prime}N^{\prime}p^{\prime}{\tau_{0}\left\lbrack {\hat{k}}_{d} \right\rbrack}} + \frac{{k_{d}_{\mu}} - {{\hat{k}}_{d}_{D}}}{2}} \right)} \times {\mathcal{F}_{m,N}^{d}\begin{bmatrix}{X_{m}{\exp \left( {- \frac{\left( {{\tau_{0}\left\lbrack {\hat{k}}_{d} \right\rbrack} + {\left( {\rho - m} \right)M}} \right)^{2}}{2s_{f}^{2}}} \right)}\mathcal{F}_{m^{\prime},N^{\prime}}^{{- 1},d}} \\\begin{matrix}{\begin{bmatrix}{X_{m^{\prime}}^{\prime}\mathcal{F}_{n^{\prime},N^{\prime}}^{{- 1},d}} \\{\left\lbrack {X_{n^{\prime}}^{\prime,*} \times {\exp \left( {- \frac{\left( {{v_{0}\left\lbrack _{\mu} \right\rbrack} + {\left( {n^{\prime} - m^{\prime}} \right)M^{\prime}}} \right)^{2}}{2s_{f}^{2}}} \right)}} \right\rbrack_{n^{\prime}}\left( {x_{1}\left( {- m} \right)} \right)_{n^{\prime}}}\end{bmatrix}_{m^{\prime}} \times} \\\left( {x_{1}(\rho)} \right)_{m^{\prime}}\end{matrix}\end{bmatrix}}_{m}\left( \frac{{Mv}_{0}\left\lbrack _{\mu} \right\rbrack}{2} \right)_{m}}},{{x_{1}(\rho)} = \frac{M^{\prime}\left( {{\tau_{0}\left\lbrack {\hat{k}}_{d} \right\rbrack} + {\rho \; M}} \right)}{2}},} & (52)\end{matrix}$

in which using the round bracket symbols in summations of IDFT-type andDFT-type, one can rearrange 5 components among 6 components in thesecond twiddle factor in (51)

[MF368]

$\begin{matrix}\left. \begin{matrix}\begin{matrix}{{{W^{{- \frac{1}{2}}n^{\prime}M^{\prime}{\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}}} \cdot W^{\frac{1}{2}n^{\prime}{mMM}^{\prime}}} = {W^{{- \frac{1}{2}}n^{\prime}{M^{\prime}{({{\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}} - {mM}})}}} = \left( {x_{1}\left( {- m} \right)} \right)_{n^{\prime}}}},} \\{{{W^{{- \frac{1}{2}}m^{\prime}M^{\prime}{\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}}} \cdot W^{{- \frac{1}{2}}m^{\prime}M^{\prime}\rho \; M}} = {W^{{- \frac{1}{2}}m^{\prime}{M^{\prime}{({{\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}} + {\rho \; M}})}}} = \left( {x_{1}(\rho)} \right)_{m^{\prime}}}},}\end{matrix} \\{{W^{\frac{1}{2}{mM}\; {\tau_{0}{\lbrack{\hat{k}}_{d}\rbrack}}} = \left( \frac{M\; {\tau_{0}\left\lbrack {\hat{k}}_{d} \right\rbrack}}{2} \right)_{m}},}\end{matrix} \right\} & (53)\end{matrix}$

and move its residue term W^(1/2 ρMτ0[k{circumflex over ( )}d]) of the 6components to the first twiddle factor in (51). The use of theseparability of the AF of the Gaussian gives (52).

Secondly let

[MF369]

Ψ_(ρ,{right arrow over (p)}) ⁽²⁾[

]

be the (ρ,p^(→))th estimated and received CE having the FD-template

[MF370]

X′ _(ρ′) U _(ρ′) ⁽²⁾[

,X]

with address

[MF371]

(ρ,{right arrow over (p)})

of the lattice

[MF372]

T _(c)

×F _(c)

,

defined by the FD-template CD of type 2

[MF373]

Ψ_(ρ,{right arrow over (p)}) ⁽²⁾[

]=d _({right arrow over (p)})

_(p′N′M′,−pNM) ^(f,d)

_(0,−pM) ^(f,d) X′ _(ρ′) U _(ρ′) ⁽²⁾[

,X].  (54)

Then one can get its associated FD correlator, called a type-2correlator

[MF374]

$\begin{matrix}{{C_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}\left( {k_{\sigma};{\hat{}}_{D}} \right)} = {\langle\begin{matrix}{{{Ae}^{i\; \kappa}W^{k_{d}_{c}}_{_{D},{- k_{d}}}^{f,d}{\Psi \left\lbrack {;\chi} \right\rbrack}},} \\\left. {W^{k_{\sigma}_{c}} \times _{{\hat{}}_{D},{- k_{\sigma}}}^{f,d}{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}\lbrack \rbrack}} \right|_{{d_{p} = 1},{X->Y},{X_{\rho^{\prime}}^{\prime}->Y_{\rho^{\prime}}^{\prime}}}\end{matrix}\rangle}_{,d}} & (55)\end{matrix}$

and have

[MF375]

$\begin{matrix}{{C_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}\left( {k_{\sigma};{\hat{}}_{D}} \right)} = {{{Ae}^{i\; \kappa}W^{{({k_{d} - k_{\sigma}})}_{c}}{\sum\limits_{ \in {\mathbb{Z}}}{_{_{D},{- k_{d}}}^{f,d}{\Psi \left\lbrack {;\chi} \right\rbrack}{{Y_{\rho}^{\prime*}\left( {_{{\hat{}}_{D},{- k_{\sigma}}}^{f,d}_{{p^{\prime}N^{\prime}M^{\prime}},{- {pNM}}}^{f,d}_{{{- \rho^{\prime}}M^{\prime}},0}^{f,d}{U_{\rho^{\prime}}^{(2)}\left\lbrack {\text{:}Y} \right\rbrack}} \right)}^{*} \cdot}}}} = {{Ae}^{j\; \kappa}W^{{({k_{d} - k_{\sigma}})}_{c}}{\sum\limits_{\overset{\rightarrow}{q}}{d_{\overset{\rightarrow}{q}}\frac{Y_{\rho^{\prime}}^{\prime,*}}{N\sqrt{N^{\prime}}}{\sum\limits_{m,m^{\prime},{n = 0}}^{{N - 1},{N^{\prime} - 1},{N - 1}}{X_{m}X_{m^{\prime}}^{\prime}Y_{n}^{*} \times {\Theta_{ZZ}\left\lbrack {{{\left( {p^{\prime} - q^{\prime}} \right){MN}} + {\hat{}}_{D} - _{D} + {\left( {n^{\prime} - m^{\prime}} \right)M^{\prime}}},{{{- \left( {p - q} \right)}{MN}} - \left( {k_{\sigma} - k_{d}} \right) - {\left( {n - m} \right)M}}} \right\rbrack} \times W^{\frac{1}{2}{({{k_{d}{\hat{}}_{D}} - {k_{\sigma}_{D}} + {2{{MNqv}_{0}{\lbrack{\hat{}}_{D}\rbrack}}} - {2M^{\prime}N^{\prime}q^{\prime}{\tau_{0}{\lbrack k_{\sigma}\rbrack}}}})}} \times {W^{\frac{1}{2}{({{{({{m\; \rho^{\prime}} - {m^{\prime}n}})}{MM}^{\prime}} - {{({m^{\prime} + \rho^{\prime}})}M^{\prime}{\tau_{0}{\lbrack k_{\sigma}\rbrack}}} + {{({m + n})}{{Mv}_{0}{\lbrack{\hat{}}_{D}\rbrack}}}})}}.}}}}}}}} & (56)\end{matrix}$

Let Y=X, Y′=X′ and discard all the terms

[MF376]

d _({right arrow over (q)})

except the term satisfying

[MF377]

{right arrow over (q)}={right arrow over (p)}.

Then one can have

[MF378]

$\begin{matrix}{{{C_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}\left( {k_{\sigma};{\hat{}}_{D}} \right)} \simeq {\frac{{Ae}^{i\; \kappa}W^{{- {\tau_{0}{\lbrack k_{\sigma}\rbrack}}}_{c}}}{\sqrt{{PP}^{\prime}}}d_{\overset{\rightarrow}{p}}X_{\rho^{\prime}}^{\prime,*}{W\left( {{{MNpv}_{0}\left\lbrack {\hat{}}_{D} \right\rbrack} - {\left( {{N^{\prime}p^{\prime}} + \frac{\rho^{\prime}}{2}} \right)M^{\prime}{\tau_{0}\left\lbrack k_{\sigma} \right\rbrack}} + \frac{{k_{d}{\hat{}}_{D}} - {k_{\sigma}_{D}}}{2}} \right)} \times {\mathcal{F}_{m^{\prime},N^{\prime}}^{{- 1},d}\begin{bmatrix}\begin{matrix}{X_{m^{\prime}}^{\prime}{\exp \left( {- \frac{\left( {{v_{0}\left\lbrack {\hat{}}_{D} \right\rbrack} + {\left( {\rho^{\prime} - m^{\prime}} \right)M^{\prime}}} \right)^{2}}{2s_{t}^{2}}} \right)}} \\{{\mathcal{F}_{m,N}^{d}\begin{bmatrix}{X_{m}{\mathcal{F}_{n,N}^{d}\left\lbrack {X_{n}^{*} \times {\exp \left( {- \frac{\left( {{\tau_{0}\left\lbrack k_{\sigma} \right\rbrack} + {\left( {n - m} \right)M}} \right)^{2}}{2s_{t}^{2}}} \right)}} \right\rbrack}_{n}} \\\left( {x_{2}\left( {- m^{\prime}} \right)} \right)_{N}\end{bmatrix}}_{m} \times}\end{matrix} \\\left( {x_{2}\left( \rho^{\prime} \right)} \right)_{m}\end{bmatrix}}_{m^{\prime}}\left( \frac{M^{\prime}{\tau_{0}\left\lbrack k_{\sigma} \right\rbrack}}{2} \right)_{m^{\prime}}}},{{x_{2}\left( \rho^{\prime} \right)} = \frac{M\left( {{v_{0}\left\lbrack {\hat{}}_{D} \right\rbrack} + {\rho^{\prime}M^{\prime}}} \right)}{2}},} & (57)\end{matrix}$

in which using the round bracket symbols in summations of IDFT-type andDFT-type, one can rearrange 5 components among 6 components in thesecond twiddle factor in (56)

[MF379]

$\begin{matrix}\left. \begin{matrix}\begin{matrix}{{{W^{\frac{1}{2}{{nMv}_{0}{\lbrack{\hat{}}_{D}\rbrack}}} \cdot W^{{- \frac{1}{2}}n\; m^{\prime}{MM}^{\prime}}} = {W^{\frac{1}{2}{{nM}{({{v_{0}{\lbrack{\hat{}}_{D}\rbrack}} - {m^{\prime}M^{\prime}}})}}} = \left( {x_{2}\left( {- m^{\prime}} \right)} \right)_{n}}},} \\{{{W^{\frac{1}{2}{mM}\; {v_{0}{\lbrack{\hat{}}_{D}\rbrack}}} \cdot W^{\frac{1}{2}{mP}\; \rho^{\prime}M^{\prime}}} = {W^{\frac{1}{2}{{nM}{({{v_{0}{\lbrack{\hat{}}_{D}\rbrack}} + {\rho^{\prime}M^{\prime}}})}}} = \left( {x_{2}\left( \rho^{\prime} \right)} \right)_{m}}},}\end{matrix} \\{{W^{{- \frac{1}{2}}m^{\prime}M^{\prime}{\tau_{0}{\lbrack k_{\sigma}\rbrack}}} = \left( \frac{M^{\prime}\; {\tau_{0}\left\lbrack k_{\sigma} \right\rbrack}}{2} \right)_{m^{\prime}}},}\end{matrix} \right\} & (58)\end{matrix}$

and move its residue term W^(−1/2 ρ′M′τ0[k∧d]) of the 6 components tothefirst twiddle factor in (56). The use of the separability of the AF ofthe Gaussian gives (57).

<6.2 Phase-Updating Loop and Von Neumann's Alternative ProjectionTheorem>

If one can get precise estimates

[MF380]

{circumflex over (k)} _(d) and

_(D)

that make all

[MF381]

|ν₀[

_(μ)]|,|τ₀[{circumflex over (k)} _(d)]|

in Lemma 2 and [MF382]

|ν₀[

_(D)]|,|τ₀[k _(σ)]|

in Lemma 4 are within chip-pulse bandwidth and duration, then the twoCCFs

[MF383]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d)) and C _(ρ,{right arrow over (p)})⁽⁴⁾(k _(σ);

_(D))

will filter out the interference

[MF384]

<ψ_(η,ρ′,{right arrow over (p)}) ⁽³⁾[k],ψ_(ρ′,{right arrow over (p)})⁽³⁾[k]>_(d,k) and <Ψ_(η,ρ,{right arrow over (p)}) ⁽⁴⁾[

],Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾[

]>_(d,l),

respectively contained in

[MF385]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d)) and C _(ρ,{right arrow over (p)})⁽⁴⁾(k _(σ):

_(D))

and recover

[MF386]

d _({right arrow over (p)}),

in place of using conventional sharp filters. This is a radical solutionto the digital signal processing for communication.The simple way to update estimates

[MF387]

{circumflex over (k)} _(d) and

_(D)

in both pairs of correlators is the following algorithm, called aPhase-Updating Loop (PUL), different from a conventional “Phase-LockedLoop” for synchronisation in communication systems:

[PUL Algorithm with Updating Rule for Attenuation Factor-MLE]:

Let [MF388]

{c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ,{circumflex over (k)} _(d,s))}_(ρ′=0) ^(N′−1)

and

[MF389]

{C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k _(σ),

_(D,s))}_(ρ=0) ^(N−1)

be the CP of the type-3 and type-4 correlator arrays, and let

[MF390]

{c _(ρ,{right arrow over (p)}) ⁽¹⁾(

_(μ) ,{circumflex over (k)} _(d,s))}_(ρ=0) ^(N−1)

and

[MF391]

{C _(ρ′,{right arrow over (p)}) ⁽²⁾(k _(σ),

_(D,s))}_(ρ′=0) ^(N′−1)

be the OP of the type-1 and type-2 correlator arrays. For simplicity set

[MF392]

d _({right arrow over (p)})=1

until the PUL algorithm is terminated. Let

[MF393]

A(θ̂_(s)^(′, d))e^(i κ(θ̂_(s)^(′, d)))

be the s-th MLE of the attenuation factor

[MF394]

A(θ^(′, d))e^(i κ(θ^(′, d)))

with the s-th digitized estimate

[MF395]

{circumflex over (θ)}′_(s) ^(,d)=({circumflex over (k)} _(d,s),

_(D,s+1)) or ({circumflex over (k)} _(d,s+1),

_(D,s))

of the parameter θ′=(t_(d), f_(D)) that is defined in (16), where

[MF396]

ψ*(t−t _(d))e ^(−i2πf) ^(D) ^((t−t) ^(d) ⁾,ψ_(w)(t)

are replaced by

[MF397]

(

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾[k])*,

ψ[k],

respectively, given as

[MF398]

$\begin{matrix}{{{\hat{A}\left( \theta_{s}^{\prime,d} \right)}e^{i\; {\hat{\kappa}{(\theta_{s}^{\prime,d})}}}} = {\frac{\sum\limits_{k = 0}^{N_{T} - 1}{\left( {_{{\hat{k}}_{d,s},{\hat{}}_{D,s}}^{d}{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\lbrack k\rbrack}} \right)^{*}_{{\hat{k}}_{d,s},{\hat{}}_{D,s}}^{d}{\psi \lbrack k\rbrack}}}{\sum\limits_{k = 0}^{N_{T} - 1}{{_{{\hat{k}}_{d,s},{\hat{}}_{D,s}}^{d}{\psi_{{\rho^{\prime},\overset{\rightarrow}{p}}\;}^{(3)}\lbrack k\rbrack}}}^{2}}.}} & (59)\end{matrix}$

Then one can update an integer pair

[MF399]

({circumflex over (k)} _(d,s),

_(D,s+1))(resp. ({circumflex over (k)} _(d,s+1),

_(D,s)))

as

{circumflex over ( )}_(D,s+1)=

*_(μ), if s is even (resp. odd), and k{circumflex over( )}_(d,s+1)=k*_(σ), if s is odd (resp. even),where

[MF400]

$\begin{matrix}{{\left. \begin{matrix}{{\left( {\rho^{\prime,*},_{\mu}^{*}} \right) = {{argmax}_{\rho^{\prime},_{\mu}}\frac{\sqrt{{PP}^{\prime}}c_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\left( {_{\mu},{\hat{k}}_{d,s}} \right)}{X_{\rho^{\prime}}^{\prime}{\hat{A}\left( \theta_{s}^{\prime,d} \right)}e^{i\; {\hat{\kappa}{(\theta_{s}^{\prime,d})}}}}}},} \\{{\left( {\rho^{*},k_{\sigma}^{*}} \right) = {{argmax}_{\rho,k_{\sigma}}\frac{\sqrt{{PP}^{\prime}}{C_{p^{\prime},\overset{\rightarrow}{p}}^{(4)}\left( {k_{\mu},{\hat{}}_{D,s}} \right)}}{X_{\rho}{\hat{A}\left( \theta_{s}^{\prime,d} \right)}e^{i\; {\hat{\kappa}{(\theta_{s}^{\prime,d})}}}}}},}\end{matrix} \right\} \mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {CP}},} & (60) \\{{\left. \begin{matrix}{{\left( {\rho^{*},_{\mu}^{*}} \right) = {{argmax}_{\rho,_{\mu}}\frac{\sqrt{{PP}^{\prime}}c_{\rho^{\prime},\overset{\rightarrow}{p}}^{(1)}\left( {_{\mu},{\hat{k}}_{d,s}} \right)}{X_{\rho}{\hat{A}\left( \theta_{s}^{\prime,d} \right)}e^{i\; {\hat{\kappa}{(\theta_{s}^{\prime,d})}}}}}},} \\{{\left( {\rho^{*},k_{\sigma}^{*}} \right) = {{argmax}_{\rho^{\prime},k_{\sigma}}\frac{\sqrt{{PP}^{\prime}}{C_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}\left( {k_{\sigma},{\hat{}}_{D,s}} \right)}}{X_{\rho^{\prime}}^{\prime}{A\left( \theta_{s}^{\prime,d} \right)}e^{i\; {\hat{\kappa}{(\theta_{s}^{\prime,d})}}}}}},}\end{matrix} \right\} \mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {OP}},} & (61)\end{matrix}$

and the initial value

[MF401]

θ′₀ ^(,d)=({circumflex over (k)} _(d,0)

_(D,1)) or ({circumflex over (k)} _(d,1),

_(D,0))

is chosen arbitrarily, e.g.,

[MF402]

θ′₀ ^(,d)=(0,0).

Select [MF403]

*_(μ) and k* _(σ)

to be candidates of

[MF404]

_(D,s+1) and {circumflex over (k)} _(d,s+1).

The (s+1)-th step estimation procedure is terminated if, for the chippulse g[k] (or G[l]) duration LΔt and bandwidth LΔf,

[MF405]

${{{\hat{k}}_{d,{s + 2}} - {\hat{k}}_{d,s}}} \leq {\frac{L}{2}\mspace{14mu} {and}\mspace{14mu} {{{\hat{}}_{D,{s + 2}} - {\hat{}}_{D,s}}}} \leq {\frac{L}{2}.}$

Then such estimates are ML estimates and both of the two correlatorsbecome ML receivers. This algorithm can be written in the form ofYoula's restoration[22] of a signal as given in Theorem below.

The recovery or restoration of a signal that has been distorted is oneof the most important problems in signal processing. Youla[22] gave theanswers to the restoration problems. Adroit use[23] of some of themathematical machinery, introduced by Youla[22] will lead us to showthat the convergence of the PUL algorithm hinges on how well the vonNeumann's APT[21] does work.

Consider the Hilbert space

[MF406]

,

consisting of all

²(Z) space of square-summable continuous and discrete-time (ordiscrete-frequency) functions with an IP, defined by

[MF407]

${\langle{f,g}\rangle}_{d,k} = {\overset{\infty}{\sum\limits_{k = {- \infty}}}{{f\lbrack k\rbrack}{g^{*}\lbrack k\rbrack}}}$

(or

[MF408]

$\left. {{{\langle{F,G}\rangle}_{d,} = {\sum\limits_{ = {- \infty}}^{\infty}{{F\lbrack \rbrack}{G^{*}\lbrack \rbrack}}}},{{F\lbrack \rbrack} = {\mathcal{F}\left\lbrack {f\lbrack k\rbrack} \right\rbrack}},{{G\lbrack \rbrack} = {\mathcal{F}\left\lbrack {g\lbrack k\rbrack} \right\rbrack}}} \right)$

and the norm

[MF409]

∥f∥=√{square root over (<f,f> _(d,k))}

or

[MF410]

∥F∥=√{square root over (<F,

)}

Let ε be any closed linear manifold (CLM) in the Hilbert space

[MF411]

.

The projection theorem[22] tells us that if ε′ and ε″ are orthogonalsub-spaces of

[MF412]

,

then every f∈ε possesses a unique decomposition f=g+h, g∈ε′, h∈ε″, whereg, h are projections of f onto ε′ and ε″, respectively, denoted by g=Pf,h=Qf; P denotes the associated projection operator (PO) projecting ontoε′, Q=I−P its associated PO projecting onto ε″, and I the identityoperator.

Let ε₁ (resp. ε₃) be the set of all

[MF413]

f∈

₂(

)

that are LΔt-TL (resp. T_(s)-TL) signals. On the contrary, denote theset of all

[MF414]

f∈

₂(

)

that are LΔf-BL signals by ε₂ and that of F_(s)-BL signals by ε₄. Eachof

[MF415]

ε_(i), 1≤i≤4

is a CLM[22].

Let P_(i) be a PO projecting onto

[MF416]

ε_(i), 1≤i≤4.

and Q_(i)=I−P_(i) the PO projecting onto the orthogonal complement ofε_(i), written as

[MF417]

⊥ε_(i), 1≤i≤4,

each of which is defined below. The CCF plays a role of a PO in thesense that given any two signals r and s, the signal r∈ε has a uniquedecomposition of the form

[MF418]

${R = {{\frac{{\langle{r,s}\rangle}_{t}}{{s}^{2}}s} + s^{\bot}}},{s \in \mathcal{E}^{\prime}},$

where

[MF419]

s ^(⊥)

is orthogonal to s. The coefficient

[MF420]

$\frac{{\langle{r,s}\rangle}_{t}}{{s}^{2}}$

is regarded as a PO projecting onto ε′.

The CP of type-3, type-4 correlators and the OP of type-1, type-2correlators provide also POs written as:

[MF421]

$\begin{matrix}{{\left. \begin{matrix}\begin{matrix}{{\psi = {{P_{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}}{\psi\psi}_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}} + \psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{{(3)}\bot}}},{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)} \in \mathcal{E}_{3}},} \\{{{P_{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}}\psi} = \frac{{\langle{\psi,{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\left( {_{\mu};{\hat{k}}_{d}} \right)}}\rangle}_{d,k}}{{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}}^{2}}},}\end{matrix} \\\begin{matrix}{{\Psi = {{P_{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(4)}}{\Psi\Psi}_{\rho^{\prime},\overset{\rightarrow}{p}}^{(4)}} + \Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{{(4)}\bot}}},{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(4)} \in \mathcal{E}_{4}},} \\{{{P_{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(4)}}\Psi} = \frac{{\langle{\Psi,{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(4)}\left\lbrack {k_{\sigma};{\hat{}}_{D}} \right)}}\rangle}_{d,}}{{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(4)}}^{2}}},}\end{matrix}\end{matrix} \right\} \mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {CP}},} & (62) \\{{\left. \begin{matrix}\begin{matrix}{{\psi = {{P_{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(1)}}{\psi\psi}_{\rho^{\prime},\overset{\rightarrow}{p}}^{(1)}} + \psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{{(1)}\bot}}},{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(1)} \in \mathcal{E}_{1}},} \\{{{P_{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(1)}}\psi} = \frac{{\langle{\psi,{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(1)}\left( {_{\mu};{\hat{k}}_{d}} \right)}}\rangle}_{d,k}}{{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}}^{2}}},}\end{matrix} \\\begin{matrix}{{\Psi = {{P_{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}}{\Psi\Psi}_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}} + \Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{{(2)}\bot}}},{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)} \in \mathcal{E}_{2}},} \\{{{P_{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}}\Psi} = \frac{{\langle{\Psi,{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}\left( {k_{\sigma};{\hat{}}_{D}} \right)}}\rangle}_{d,}}{{\Psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(2)}}^{2}}},}\end{matrix}\end{matrix} \right\} \mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {OP}},} & (63)\end{matrix}$

where

[MF422]

ψ_(ρ,{right arrow over (ρ)}) ^((1)⊥),Ψ_(ρ′,{right arrow over (p)})^((2)⊥),ψ_(ρ′,{right arrow over (p)})^((3)⊥),Ψ_(ρ,{right arrow over (p)}) ^((4)⊥)

are orthogonal complements of

[MF423]

ψ_(ρ,{right arrow over (ρ)}) ⁽¹⁾,Ψ_(ρ′,{right arrow over (p)})⁽²⁾,ψ_(ρ′,{right arrow over (p)}) ⁽³⁾,Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾.

The Alternative Projection Theorem (APT) [21, p. 55, theorem 13.7] (FIG.11)

tells us that:If E and F are projections on CLMs

[MF424]

ε,

in a Hilbert space, respectively, then the sequence of operators E, FE,EFE, FEFE, . . . has a limit G, the sequence F, EF, FEF, . . . has thesame limit G, and G is a projection on

[MF425]

ε

.

(The condition EF=FE need not hold.)With use of APT, the following result is obtained:

Theorem: Convergence of Phase Updating Loop (PUL) (see FIG. 3 c):

Let us consider the 4 POs in the TD and FD that contain (s+1)-stepestimates

[MF426]

(ρ′^(,)*,

*_(μ))(resp.(ρ*,

*_(μ))),

determined by the argmax-operations in terms of (ρ′,

_(μ)) (resp. (ρ,

_(μ))) with the s-step estimate

[MF427]

{circumflex over (k)} _(ds)

in (60) (resp. (61)), and (ρ*,k*_(σ)) (resp. (ρ″*,k*_(σ))), determinedby the argmax-operations in terms of (ρ,k_(σ)) (resp. (ρ′,k_(σ))) withthe s-step estimate

[MF428]

_(Ds)

in (60) (resp. (61)). Write these POs symbolically as

[MF429]

$\begin{matrix}\left. \begin{matrix}{{P_{3} = {P_{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}}\left( {\rho^{\prime},{_{\mu};{\hat{k}}_{ds}}} \right)}},{P_{4} = {{P_{\Psi_{\rho,\overset{\rightarrow}{p}}^{(4)}}\left( {\rho,{k_{`\sigma};{\hat{}}_{Ds}}} \right)}\mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {CP}}},} \\{{P_{1} = {P_{\psi_{\rho^{\prime},\overset{\rightarrow}{p}}^{(1)}}\left( {\rho,{_{\mu};{\hat{k}}_{ds}}} \right)}},{P_{2} = {{P_{\Psi_{\rho^{\prime`},\overset{\rightarrow}{p}}^{(2)}}\left( {\rho^{\prime},{k_{\sigma};{\hat{}}_{Ds}}} \right)}\mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {OP}}},}\end{matrix} \right\} & (64)\end{matrix}$

and

[MF430]

Q _(k) =I−P _(k), 1≤k≤4.

Then the PUL algorithm converges.

[Proof]: Consider the recursion formula for the CP only because that forthe OP is exactly the same except an exchange between the TD-PC X andFD-PC X′, i.e., between suffixes (3, 4) and (1, 2). Two differentorderings of the time-truncation operator P₃ and the band-limitedoperator P₄ provide two different iteration equations;

First consider an algorithm for the recovery of

[MF431]

ψ

as follows: If

[MF432]

ψ∈ε₄,

then

[MF433]

ψ=

^(−1,d) P ₄

^(d)ψ

and

[MF434]

g ₁ =P ₃ ψ=P ₃

^(−1,d) P ₄

^(d) ψ=ψ−Q ₃

^(−1,d) P ₄

^(d)ψ,

Thus [MF435]

ψ

satisfies the operator equation

[MF436]

A ₁ ψ=g ₁ , A ₁ =I−Q ₃

^(−1,d) P ₄

^(d).

The recursion, defined by

[MF437]

ψ=g ₁ +Q ₃

^(−1,d) P ₄

^(d)ψ

enables us to have the iterative equation in the TD[22, 23]

[MF438]

ψ_(i)=(I−Q ₃

^(−1,d) P ₄

^(d))ψ+Q ₃

^(−1,d) P ₄

^(d)ψ_(i−1),

or ψ_(i)−ψ=(Q ₃

^(−1,d) P ₄

^(d))^(i)(ψ₀−ψ), if (ψ₀−ψ)∈ε₄.  (65)

By the APT, [MF439]

$\lim\limits_{i\rightarrow\infty}{\left( {Q_{3}\mathcal{F}^{{- 1},d}P_{4}\mathcal{F}^{d}} \right)^{i}\left( {\psi_{0} - \psi} \right)}$

becomes the projection of

[MF440]

(ψ₀−ψ)

onto a CLM ε_(c)=⊥ε₃∩ε₄. The ε_(c) contains only the trivialfunction[22, p. 699], [23, p. 637]; i.e.,

[MF441]

${\lim\limits_{i\rightarrow\infty}\psi_{i}} = {\psi.}$

This is one of Youla's[22] main results. Hence the operator

[MF442]

P ₃

^(−1,d) P ₄

^(d)

singles out the rectangle of chip-pulse duration LΔt and chip bandwidthLΔf with chip-level and data-level addresses

[MF443]

((ρ,ρ′),{right arrow over (p)}),

i.e., the intersection of the rectangular support of its associated CE

[MF444]

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾

and that of the CE

[MF445]

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾,

i.e.,

[MF446]

[((p−1)N+ρ−Δa′)T _(c),((p−1)N+ρ+Δb′)T _(c)]×[((p′−1)N′+ρ′−Δa)F_(c),((p′−1N′+ρ′+Δb)F _(c)]

of the lattice

[MF447]

T _(c)

×F _(c)

and filters out the rest in the TFP, where Δa′, Δb′, Δa, and Δb areintegers satisfying Δa′+Δb′=M′ and Δa+Δb=M. Such an operator is referredto as a phase-space (or time-frequency) localization operator[7]. Thusthe CE ψ[k] is restored so that the parameters

[MF448]

k _(d) and

_(D)

are estimated within LΔt and LΔf. It should be noted that unlike usualrectangles of strictly TL (resp. BL) operator[23], the time-truncationoperator P₃ (resp. the band-limited operator P₄) is defined by using atemplate, i.e., a non-overlapped superposition of N coded TD Gaussianchip-pulses g[k] (resp. N′ coded FD Gaussian chip-pulses

[MF449]

G(

)

with no guard interval.

Conversely, in the FD, if ψ∈ε₃, then

[MF450]

Ψ=

^(d) P ₃

^(−1,d)Ψ

and

[MF451]

g ₂ =P ₄ Ψ=P ₄

^(d) P ₃

^(−1,d) Ψ=Ψ−Q ₄

^(d) P ₃

^(−1,d)Ψ.

That is, ψ satisfies the operator equation [23]

[MF452]

A ₂ Ψ=g ₂ , A ₂ =I−Q ₄

^(d) P ₃

^(−1,d).

This gives the iterative equation

[MF453]

Ψ_(i)=(I−Q ₄

^(d) P ₃

^(−1,d))Ψ+Q ₄

^(d) P ₃

^(−1,d)Ψ_(i−1),

or Ψ_(i)−Ψ=(Q _(i)

^(d) P ₃

^(−1,d))^(i)(Ψ₀−Ψ), if (Ψ₀−Ψ)∈ε₃.  (66)

By the APT, [MF454]

$\lim\limits_{i\rightarrow\infty}{\left( {Q_{4}\mathcal{F}^{d}P_{3}\mathcal{F}^{{- 1},d}} \right)^{i}\left( {\Psi_{0} - \Psi} \right)}$

becomes the projection of (ψ₀-ψ) onto a CLM

[MF455]

ε′_(c)=⊥ε₄∩ε₃.

This CLM contains only the trivial function. Thus lim_(i→∞)ψ_(i)=ψ.

The FD CE

[MF456]

Ψ[

]

is restored so that the parameters

[MF457]

k _(d) and

_(D)

are estimated within LΔt and LΔf. Similarly, the operator

[MF458]

P ₄

^(d) P ₃

^(−1,d),

i.e., another localization operator singles out the rectangle of LΔt×LΔfwith chip-level and data-level addresses

[MF459]

((ρ,ρ′),{right arrow over (p)})

i.e., the intersection of the rectangular support of the CE

[MF460]

ψ_(ρ′,{right arrow over (p)}) ⁽³⁾

and that of the CE

[MF461]

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾,

[MF462]

[((p−1)N+ρ−Δa′)T _(c),((p−1)N+ρ+Δb′)T _(c)]×[((p′−1)N′+ρ′−Δa)F_(c),((p′−1)N′+ρ′+Δb)F _(c)]

of the lattice

[MF463]

T _(c)

×F _(c)

and filters out the rest in the TFP. [End of proof]

<7. Twinned Filter Bank Multi-Carrier System> <7.1 Transmitters forGenerating Signature and Radar Signal>

Consider a multi-carrier filter bank for generating the TD-signaturev[k;χ] and FD-signature

[MF464]

V[

;χ],

defined in (25). Using a repetition of the TD-PC X_(m) of period N toget an infinite code sequence at the time: −∞, . . . , −T_(c), 0, T_(c),. . . , ∞ and that of the FD-PC X′_(m′) of period N′ to get an infinitecode sequence at the frequency: −∞, . . . , −F_(c), 0, F_(c), . . . , ∞,one can get: Proposition 6: Write v[k;χ] and

[MF465]

V[

;χ]

in (25) respectively as

[MF466]

$\begin{matrix}\left. \begin{matrix}{{{v\left\lbrack {k;\chi} \right\rbrack} = {\frac{1}{\sqrt{{NN}^{\prime}}}{\sum\limits_{m^{\prime} = 0}^{N^{\prime} - 1}{X_{m^{\prime}}^{\prime}{\sum\limits_{m \in {\mathbb{Z}}}{X_{m}{f_{m,m^{\prime}}^{TD}\left\lbrack {k - {mM}} \right\rbrack}}}}}}},} \\{{{V\left\lbrack {;\chi} \right\rbrack} = {\frac{1}{\sqrt{{NN}^{\prime}}}{\sum\limits_{m = 0}^{N - 1}{X_{m}{\sum\limits_{m^{\prime} \in {\mathbb{Z}}}{X_{m^{\prime}}^{\prime}{f_{m,m^{\prime}}^{FD}\left\lbrack { - {m^{\prime}M^{\prime}}} \right\rbrack}}}}}}},}\end{matrix} \right\} & (67)\end{matrix}$

where

[MF467]

f _(m,m′) ^(TD),[k],f _(m,m′) ^(FD)[

]

are modulated filters, defined by

[MF468]

$\begin{matrix}\left. \begin{matrix}{{{f_{m,m^{\prime}}^{TD}\lbrack k\rbrack} = {{{g\lbrack k\rbrack}W^{{- m^{\prime}}{M^{\prime}{({k + \frac{mM}{3}})}}}} = {{g\lbrack k\rbrack}{W^{{- m^{\prime}}M^{\prime}k}\left( {- 1} \right)}^{m\; m^{\prime}}}}},} \\{{f_{m,m^{\prime}}^{FD}\lbrack \rbrack} = {{{G\lbrack \rbrack}W^{{mM}{({ + \frac{m^{\prime}M^{\prime}}{2}})}}} = {{G\lbrack \rbrack}{{W^{{mM}\; }\left( {- 1} \right)}^{m\; m^{\prime}}.}}}}\end{matrix} \right\} & (68)\end{matrix}$

For an input x[k] and output y[n], Vaidyanathan[13, p 117] defined theinput-output relation in the TD using three types of multi-rate filterswith filter coefficient h[⋅]: M_(f)-fold decimation filter:

[MF469]

${{y\lbrack n\rbrack} = {\sum\limits_{k \in {\mathbb{Z}}}{{x\lbrack k\rbrack}{h\left\lbrack {{nM}_{f} - k} \right\rbrack}}}},$

L_(f)-fold interpolation filter:

${{y\lbrack n\rbrack} = {\sum\limits_{k \in {\mathbb{Z}}}{{x\lbrack k\rbrack}{h\left\lbrack {n - {kL}_{f}} \right\rbrack}}}},$

and M_(f)/L_(f)-fold decimation filter:

[MF471]

${y\lbrack n\rbrack} = {\sum\limits_{k \in {\mathbb{Z}}}{{x\lbrack k\rbrack}{{h\left\lbrack {{nM}_{j} - {kL}_{f}} \right\rbrack}.}}}$

[Proof]: The TD-signature υ[k;χ] in (25) (resp. the FD-signature

[MF472]

V[

;χ]

) looks like the expression of a signal obtained at the output of asynthesis filter bank (SFB)[10, 12] with N′ (resp. N) sub-bands and withan expansion factor equal to M (resp. M′) and being phase-coded by theFD-PC X′_(m′) (resp. the TD-PC X_(m)) on each sub-band. Indeed, if the

[MF473]

X _(m) ,m∈

(resp. X′ _(m) ,m′∈

)

is the input signal and the

[MF474]

f _(m,m′) ^(TD)[k](resp. f _(m,m′) ^(FD)[

])

is the filter on the (m,m′)th sub-band of this SFB, then the output

[MF475]

υ[k;χ](resp. V[

;χ])

can be written as (67). [End of proof]

Together with the symbols lcm[M,N′]=M₀N′=MN′₀, lcm[M′,N]=M′₀ N=M′N₀[13],[10],[11] using M₀N′,M′₀ N polyphase component

[MF476]

$\begin{matrix}{{{E_{e}^{g}(z)} = {\sum\limits_{k \in {\mathbb{Z}}}{{g\left\lbrack {e + {{kM}_{0}N^{\prime}}} \right\rbrack}z^{- k}}}},{{E_{e}^{G}(z)} = {\sum\limits_{\; \in {\mathbb{Z}}}{{G\left\lbrack {e + {\; M_{0}^{\prime}N}} \right\rbrack}z^{- }}}}} & (69)\end{matrix}$

one can get polyphase filters (Vaidyanathan's[13, p 121] Type 1polyphase)

[MF477]

$\begin{matrix}\left. \begin{matrix}{{{F_{m\; m^{\prime}}^{TD}(z)} = {{\sum\limits_{k = 0}^{{M_{0}N^{\prime}} - 1}{{f_{m\; m^{\prime}}^{TD}\lbrack k\rbrack}z^{- k}}} = {\left( {- 1} \right)^{m\; m^{\prime}}{\sum\limits_{e = 0}^{{M_{0}N^{\prime}} - 1}{E_{e}^{g}\left( {zW}^{m^{\prime}M^{\prime}} \right)}}}}},} \\{{{F_{m\; m^{\prime}}^{FD}(z)} = {{\sum\limits_{k = 0}^{{M_{0}^{\prime}N} - 1}{{f_{m\; m^{\prime}}^{FD}\lbrack \rbrack}z^{- }}} = {\left( {- 1} \right)^{m\; m^{\prime}}{\sum\limits_{e = 0}^{{M_{0}^{\prime}N^{\prime}} - 1}{E_{e}^{G}\left( {zW}^{- {mM}} \right)}}}}},}\end{matrix} \right\} & (70)\end{matrix}$

For l_(m)=lcm[M,N′], g_(d)=gcd[M,N′], there exist N′₀,M₀ satisfyingl_(m)=MN′₀=N′M₀. On the other hand, by the identity l_(m)·g_(d)=M·N′,M=g_(d)·M₀,N′=g_(d)N′₀ hold.

Thus one can get the SFBs of 2-D modulated v[k] and V[

] by TD-, FD-PC as shown in FIGS. 4,5, respectively.

FIG. 4 shows the SFB with TD-, FD-PCs X_(m′)X′_(m′) for generating TDsignature v[k] in (25), (67), containing m′-th TD template

[MF478]

u _(m′) ^(TD)[k], 0≤m′≤N′−1.

FIG. 5 shows the SFB with TD-, FD-PCs X_(m′)X′_(m′) for generating FDsignature

[MF479]

V[

]

in (25), (67), containing m-th FD template

[MF480]

U _(m) ^(FD)[

], 0≤m≤N−1.

While eq. (27) provides the SFBs for generating

[MF481]

ψ[k;χ]

and

[MF482]

Ψ[

;χ].

Proposition 7: The CEs ψ[k;χ] and

[MF483]

Ψ[

;χ].

can be written respectively as

[MF484]

$\begin{matrix}\left. \begin{matrix}{{{\psi \left\lbrack {k;\chi} \right\rbrack} = {\frac{1}{\sqrt{{PP}^{\prime}}}{\sum\limits_{q,{q^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}{d_{\overset{\rightarrow}{q}} \cdot {k_{q,q^{\prime}}^{TD}\left\lbrack {k - {qNM}} \right\rbrack}}}}},} \\{{{\Psi \left\lbrack {;\chi} \right\rbrack} = {\frac{1}{\sqrt{{PP}^{\prime}}}{\sum\limits_{q,{q^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}{d_{\overset{\rightarrow}{q}} \cdot {k_{q,q^{\prime}}^{FD}\left\lbrack { - {q^{\prime}N^{\prime}M^{\prime}}} \right\rbrack}}}}},}\end{matrix} \right\} & (71)\end{matrix}$

where

[MF485]

k _(q,q′) ^(TD)[k],k _(q,q′) ^(FD)[

]

are modulated filters of their associated SFBs, defined by

[MF486]

$\begin{matrix}\left. \begin{matrix}{{{k_{q,q^{\prime}}^{TD}\lbrack k\rbrack} = {{{v\left\lbrack {k;\chi} \right\rbrack}W^{{- q^{\prime}}N^{\prime}{M^{\prime}{({k + \frac{qNM}{2}})}}}} = {{v\left\lbrack {k;\chi} \right\rbrack}{W^{{- q^{\prime}}N^{\prime}M^{\prime}k}\left( {- 1} \right)}^{{qq}^{\prime}{NN}^{\prime}}}}},} \\{{k_{q,q^{\prime}}^{FD}\lbrack \rbrack} = {{{V\left\lbrack {;\chi} \right\rbrack}W^{{qNM}{({ + \frac{q^{\prime}N^{\prime}M^{\prime}}{2}})}}} = {{V\left\lbrack {;\chi} \right\rbrack}{{W^{{qNM}\; }\left( {- 1} \right)}^{{qq}^{\prime}{NN}^{\prime}}.}}}}\end{matrix} \right\} & (72)\end{matrix}$

[Proof]: Equation (71) indicates that if

[MF487]

{d _(q,q′)}_(q=0) ^(P−1)

(resp.

[MF488]

{d _(q,q′)}_(q′=0) ^(P′−1))

denotes the input signal, and

[MF489]

k _(q,q′) ^(TD)[k]

(resp.

[MF490]

k _(q,q′) ^(FD)[

])

is the filter on the (q,q′)th sub-band of its associated SFB, then the

[MF491]

TD-CE:ψ[k;χ]

(resp.

[MF492]

FD-CE:Ψ[

;χ])

is the output of its SFB with PP′ sub-bands and with an expansion factorequal to NM (resp. N′M′) on each sub-band. [end of proof]

Thus one can obtain the SFB for generating a transmit signal, thatcontains data symbols as shown in FIGS. 6,7. Conventional SFBs do notinvolve the process of generating signature as shown in FIGS. 4,5. InFIGS. 6,7, the case where N=N′=1 corresponds to the usual SFBs. Usingthe following modulated filter (MF)s for data transmission, given by

[MF493]

$\begin{matrix}{{{E_{e}^{v}(z)} = {\sum\limits_{k \in {\mathbb{Z}}}{{v\left\lbrack {e + {{kM}_{0}N^{\prime}}} \right\rbrack}z^{- k}}}},{{E_{e}^{V}(z)} = {\sum\limits_{ \in {\mathbb{Z}}}{{V\left\lbrack {e + {\; M_{0}^{\prime}N}} \right\rbrack}z^{- }}}},} & (73)\end{matrix}$

one can obtain polyphase filters (Vaidyanathan's[13, p 121] Type 1polyphase)

[MF494]

$\begin{matrix}\left. \begin{matrix}{{{K_{{qq}^{\prime}}^{TD}(z)} = {{\sum\limits_{k = 0}^{{P_{0}^{\prime}{MN}} - 1}{{k_{{qq}^{\prime}}^{TD}\lbrack k\rbrack}z^{- k}}} = {\left( {- 1} \right)^{{qq}^{\prime}{NN}^{\prime}}{\sum\limits_{e = 0}^{{P_{0}^{\prime}{MN}} - 1}{E_{e}^{v}\left( {zW}^{q^{\prime}N^{\prime}M^{\prime}} \right)}}}}},} \\{{{K_{{qq}^{\prime}}^{FD}(z)} = {{\sum\limits_{ = 0}^{{P_{0}{MN}} - 1}{{k_{{qq}^{\prime}}^{FD}\lbrack \rbrack}z^{- }}} = {\left( {- 1} \right)^{{qq}^{\prime}{NN}^{\prime}}{\sum\limits_{e = 0}^{{P_{0}M^{\prime}N^{\prime}} - 1}{E_{e}^{V}\left( {zW}^{- {qNM}} \right)}}}}},}\end{matrix} \right\} & (74)\end{matrix}$

where P′₀,P₀ denote integers satisfying P′₀MN=lcm[P′,MN],P₀MN=lcm[P,MN]. [end of proof]

FIG. 6 shows the SFB for generating TD-complex envelope (CE) in (26)

[MF495]

ψ[k]

with input, complex-valued data

[MF496]

{d _(p,p′)}_(p,p′=1) ^(P,P′),

FIG. 7 shows the SFB for generating FD-CE in (26)

[MF497]

Ψ[

]

with input, complex-valued data

[MF498]

{d _(p,p′)}_(p,p′=1) ^(P,P′).

<7.2 AFB: Receivers and Encoder Design>

Besides the attenuation factors Ae^(iκ), the PD due to non-commutativityof modulation/demodulation by the carrier l_(c) operations, accompaniedby the delay k{circumflex over ( )}_(d),

[MF499]

arises. The received signal in (28)

[MF500]

r[k;χ,A,κ,θ ^(t,d)]

(resp. its FT

[MF501]

R[

;χ,A,κ,θ′ ^(,d)]),

(simply denoted by

[MF502]

r[k],R[

])

contains those PDs.Define a type-3 (resp. type-4) CCF between the received signalr[k](resp. R[l] in (28) and the estimated TD-template CE of type 3 in(29)

[MF503]

[k]|_(d) _(p) ₌₁

(resp. FD-CE in (33)

[MF504]

Ψ_(ρ,{right arrow over (p)}) ⁽⁴⁾[

]_(d) _({right arrow over (p)}) ₌₁),

respectively by

[MF505]

${{c_{\rho^{\prime},\overset{\_}{p}}^{{(3)},s}\left( {_{\mu};{\hat{k}}_{d}} \right)} = {\sum\limits_{k \in {\mathbb{Z}}}{{r\lbrack k\rbrack}\left( {W^{{\hat{k}}_{d}_{c}}_{{\hat{k}}_{d},_{\mu}}^{d}_{{pNM};{p^{\prime}N^{\prime}M^{\prime}}}^{d}_{0,{p^{\prime}M^{\prime}}}^{d}X_{p^{\prime}}^{\prime}{u_{p^{\prime}}^{(3)}\left\lbrack {k;X} \right\rbrack}} \right)^{*}}}},{{C_{\rho,\overset{\_}{p}}^{{(4)},R}\left( {k_{\sigma};{\hat{}}_{D}} \right)} = {\sum\limits_{ \in {\mathbb{Z}}}{{R\lbrack \rbrack}\left( {W^{k_{\sigma}_{c}}_{{\hat{}}_{D},{- k_{\sigma}}}^{f,\; d}_{{p^{\prime}N^{\prime}M^{\prime}},{- {pNM}}}^{f,\; d}_{0,{- {\rho M}}}^{f,\; d}X_{\rho}{U_{\rho}^{(4)}\left\lbrack {\text{:}X^{1}} \right\rbrack}} \right)^{*}}}},$

where the phase factor

[MF506]

(resp.

[MF507]

)

is designed to cancel out the one

[MF508]

in the signal component of r[k] in (28), (38) (cf.

[MF509]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d)))

(resp. that of

[MF510]

R[

])

(cf. in (28), (43)

[MF511]

C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k _(σ);

_(D))).

Then one can get a TD (resp. FD) analysis filter bank (AFB)[10, 12]:

Proposition 8: The CCFs of type-3 and of type-4 can be rewritten, suitedto AFB realizations, respectively as

[MF512]

$\begin{matrix}{{{c_{\rho^{\prime},\overset{\_}{p}}^{{(3)},x}\left( {_{\mu};{\hat{k}}_{d}} \right)} = {\sum\limits_{k \in {\mathbb{Z}}}{{r\left\lbrack {k + {\hat{k}}_{d}} \right\rbrack}W^{{- {\hat{k}}_{d}}_{c}}W^{{{_{p}{\lbrack{k - \frac{{pNM} - {\hat{k}}_{d}}{2}}\rbrack}}{h_{\rho^{\prime},\overset{\_}{p}}^{(3)}{\lbrack{{pNM} - k + D}\rbrack}}},}}}}} & (75) \\{{{C_{\rho^{\prime},\overset{\_}{p}}^{{(4)},R}\left( {k_{\sigma};{\hat{}}_{D}} \right)} = {\sum\limits_{ \in {\mathbb{Z}}}{{R\left\lbrack { + {\hat{}}_{D}} \right\rbrack}W^{{- k_{\sigma}}_{c}}W^{- {k_{\sigma}\lbrack{ - \frac{{{p^{\prime}N^{\prime}M^{\prime}} - _{x}},}{2}})}}{h_{p,\overset{\_}{p}}^{(4)}\left\lbrack {{p^{\prime}N^{\prime}M^{\prime}} - } \right\rbrack}}}},} & (76)\end{matrix}$

where D=L−1 and

[MF513]

h _(ρ′,{right arrow over (p)}) ⁽³⁾[k],h _(ρ,{right arrow over (p)}) ⁽⁴⁾[

]

are modulated filters of TD and FD AFBs

[MF514]

$\begin{matrix}\left. \begin{matrix}{{h_{\rho^{\prime}\overset{\_}{\rho}}^{(3)}\lbrack k\rbrack} = {X_{\rho^{\prime}}^{\prime,*}{u_{\rho^{\prime}}^{{(3)},*}\left\lbrack {k;X} \right\rbrack}W^{{{({{\rho^{\prime}N^{\prime}} + \rho^{\prime}})}{M^{\prime}{({k - D - \frac{{\rho \; {\mathcal{M}}} - {\hat{x}}_{d}}{2}})}}},}}} \\{{h_{\rho^{\prime}\overset{\_}{\rho}}^{(4)}\lbrack \rbrack} = {X_{\rho}^{*}{U_{\rho}^{{(4)},*}\left\lbrack {;X^{1}} \right\rbrack}W^{{({{{- {({{\rho \; N} + \rho})}}M} - D})}}W^{{({{pN} + \rho})}M{\frac{{\rho^{\prime}^{\prime}\mathcal{M}^{\prime}} + r_{D}}{2}.}}}}\end{matrix} \right\} & (77)\end{matrix}$

(When P×P′ [MF515]

-ary symbols

are transmitted,

one may evaluate the output at the address

[MF516]

{right arrow over (p)}=(p,p′)

in the correlator array of p-band, p′-duration to estimate

[MF517]

{d _({right arrow over (p)})}_(p,p′=1) ^(P,P′).)

[Proof]: If the phase-modulated received TD signal

[MF518]

$W^{{- {\hat{k}}_{d}}_{c}}W^{_{\mu}{\lbrack{k - \frac{{v\; {\mathcal{M}}} - {\hat{u}}_{d}}{2}}\rbrack}}{r\left\lbrack {k + {\hat{k}}_{d}} \right\rbrack}$

(resp. the FD one

[MF519]

$\left. {W^{{- {\hat{k}}_{\sigma}}_{c}}W^{- {k_{a}{\lbrack{ - \frac{{p^{\prime}^{\prime}\mathcal{M}^{\prime}} - {\hat{}}_{D}}{2}}\rbrack}}}{R\left\lbrack { + {\hat{}}_{D}} \right\rbrack}} \right)$

denotes the input signal of the

[MF520]

(ρ′,{right arrow over (p)})-th

(resp.

[MF521]

(ρ,{right arrow over (p)})-th)

-sub-band of the AFB with P′ (resp. P) sub-bands with a decimationfactor equal to NM (resp. N′M′) on each sub-band, given by (75) (resp.(76)), then two facts:i) one is the TD symmetry property[10] (cf. (21))

[MF522]

g(−t)=g(t), i.e., g[k]=g[D−k], u _(ρ′) ⁽³⁾[k;X]=u _(ρ′) ⁽³⁾[D−k;X]

and ii) the other is the property of the FD signal

[MF523]

G[−

]=W ^(−D)

G[

], U _(p) ⁽⁴⁾[−

;X′]=W ^(−D)

U _(ρ) ⁽⁴⁾[

;X′],

that the TD symmetry entails, prove that its associated filter

[MF524]

h _(ρ′,{right arrow over (p)}) ⁽³⁾[k]

(resp.

[MF525]

h _(ρ,{right arrow over (p)}) ⁽⁴⁾[

])

is defined by (77). [end of proof]Note that two kinds of filters given in patent[2] contain neither the PD

nor its canceling-out terms

,

.

If AFB filter in (77) and P′₀MN, P₀M′N′ polyphase components:

[MF526]

$\begin{matrix}\left. \begin{matrix}{{{R_{e}^{u_{e^{\prime}}}(z)} = {\sum\limits_{k \in {\mathbb{Z}}}{u_{s^{\prime}}^{TD}\left\lbrack {e + {{kP}_{0}^{\prime}{MN}}} \right\rbrack}_{z}^{- k}}},{1 \leq e \leq {P_{0}^{\prime}{MN}}},} \\{{{R_{e^{\prime}}^{U}(z)} = {\sum\limits_{ \in {\mathbb{Z}}}{U_{s}^{FD}\left\lbrack {e + {\; P_{0}M^{\prime}N^{\prime}}} \right\rbrack}_{z}^{- }}},{1 \leq e^{\prime} \leq {P_{0}M^{\prime}N^{\prime}}},}\end{matrix} \right\} & (78)\end{matrix}$

then one can obtain N′,N polyphase filters (Vaidyanathan's[13] Type 2polyphase)

[MF527]

$\begin{matrix}\left. \begin{matrix}\begin{matrix}\begin{matrix}{{H_{\overset{\_}{p},s^{\prime}}^{TD}\left( {z;{\hat{k}}_{d}} \right)} = {{\sum\limits_{k = 0}^{{P_{0}^{\prime}{MN}} - 1}{{h_{\overset{\_}{p},\; s^{\prime}}^{TD}\lbrack k\rbrack}z^{- k}}} = {\left( {- 1} \right)^{{pp}^{\prime}{NN}^{\prime}}W^{({{p^{\prime}N^{\prime}} + \; s^{\prime}})}}}} \\{{M^{\prime}\frac{R_{d}}{2}{\sum\limits_{\epsilon = 0}^{{P_{0}^{\prime}{MN}} - 1}{R_{c}^{u_{s^{\prime}}}\left( {zW}^{{({{p^{\prime}N^{\prime}} + \; s^{\prime}})}M^{\prime}} \right)}}},{1 \leq s^{\prime} \leq N^{\prime}}}\end{matrix} \\{{H_{\overset{\_}{p},s}^{FD}\left( {z;{\hat{}}_{d}} \right)} = {{\sum\limits_{k = 0}^{{P_{0}^{\prime}M^{\prime}N^{\prime}} - 1}{{h_{\overset{\_}{p},\; s}^{FD}\lbrack \rbrack}z^{- }}} = {\left( {- 1} \right)^{{pp}^{\prime}{NN}^{\prime}}W^{- {({{pN} + \; s})}}}}}\end{matrix} \\{{M\frac{r_{D}}{2}{\sum\limits_{\epsilon = 0}^{{P_{0}^{\prime}M^{\prime}N^{\prime}} - 1}{R_{c^{\prime}}^{U_{s}}\left( {zW}^{{- {({{pN} + \; s})}}M} \right)}}},{1 \leq s \leq N}}\end{matrix} \right\} & (79)\end{matrix}$

Thus one can get the AFBs as shown in FIGS. 8, 9. (The case whereN=N′=1, N₀=N′₀=1 corresponds to the usual AFBs [12, 13].) The two CCFs(75) and (76) are FBMC-realizations of c⁽³⁾ _(ρ′,p→)(

_(μ);k_(d)),C⁽⁴⁾ _(ρ,p→)(k_(σ);

_(D)), respectively;

The output bipolar symbol of the TD-AFB (resp. the FD-AFB) is given asthe sign of

[MF528]

$\frac{X_{p^{\prime}}^{\prime}{c_{p^{\prime},\overset{\_}{p}}^{{(3)},r}\left( {_{\mu};{\hat{k}}_{d}} \right)}}{\hat{A}e^{i\hat{}}}$

(resp. that of

[MF529]

$\left. {\frac{X_{p}{C_{p,\overset{¨}{p}}^{{(4)},R}\left( {k_{\sigma};{\hat{}}_{D}} \right)}}{\hat{A}e^{i\; }}} \right),$

where

[MF530]

Âe ^(i{circumflex over (κ)})

is the MLE of the attenuation factor Ae^(iκ) with the MLE

[MF531]

{circumflex over (θ)}′^(,d)=({circumflex over (k)} _(d),

_(D))

(cf. (59)). Such a pair of SFBs and AFBs symmetrical in the TD and FD isreferred to as a “twinned-FBMC”.

FIG. 8 shows an AFB equipped with a TD-correlator array for decodingcomplex-valued data symbols

[MF532]

{d _(p,p′)}_(p=1) ^(P), 1≤p′≤P′.

FIG. 9 shows the AFB equipped with an FD-correlator array for decodingcomplex-valued data symbols

[MF533]

{d _(p,p′)}_(p′=1) ^(P′), 1≤p≤P,

In which the TD-correlator

[MF534]

C _({right arrow over (p)},s′) ^(TD)(

_(μ) ;{circumflex over (k)} _(d))

and the FD-correlator [MF535]

C _({right arrow over (p)},s′) ^(FD)(k _(σ);

_(D))

in FIGS. 8 and 9 correspond respectively to

[MF536]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d))

and

[MF537]

C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k _(σ);

_(D)).

Note that if the PUL algorithm is implemented in the interface betweenthe TD- and FD-AFBs, as shown in FIG. 10, then the resultant interfacewith time-varying or adaptive AFBs becomes a parameter estimator for aradar, and a synchroniser[3] for communication systems after the PULalgorithm converges. In addition, it also plays a role of an encoder ofthe transmitted

[MF538]

-ary symbol d _({right arrow over (p)})∈

for data communication on the basis of complex-valued CCFs

[MF539]

$\frac{X_{p^{\prime}}^{\prime}{c_{p^{\prime}}^{{(3)},r}\left( {_{\mu};{\hat{k}}_{d}} \right)}}{\hat{A}e^{i\hat{}}}$

and

[MF540]

$\frac{X_{p}{C_{p,\overset{¨}{p}}^{{(4)},R}\left( {k_{\sigma};{\hat{}}_{D}} \right)}}{\hat{A}e^{i\; \hat{}}}.$

All the filters, defined by (68), (72), and (77) in this FBMC can berealized by the polyphase filters (called Vaidyanathan's type 1 or type2 polyphase[13]). But such subjects are beyond our present interest andso are omitted here.

FIGS. 10a and 10c show the AFBs with the TD-correlator array and theFD-correlator array, respectively, and FIG. 10b shows an illustration ofan alternative updating process between MLEs of two arrays based on thevon Neumann's APT.

<8. Other Examples of Communication Exploiting the Non-Commutativity ofTime and Frequency Shifts>

In the radar detection technique as a typical example of communicationexploiting the non-commutative property (NCP) of time and frequencyshift (TFS)s, a pair of TFSO of type 3 with t_(d), f_(D)

[MF541]

_(t) _(d) _(,f) _(D)

and TFSO of type 4

[MF542]

_({circumflex over (t)}) _(d) _(,{circumflex over (f)}) _(D)

with estimated parameters

[MF543]

{circumflex over (t)} _(d) ,{circumflex over (f)} _(D)

at the receiver, has shown that a priori half shifts of t_(d), f_(D)play an important role in the estimation of those parameters. Otherexamples of communication systems exploiting the NCP of TFSs are givenas follows.

Our discussion has been restricted to a single-target problem forsimplicity. Of course, a simple way for detecting multiple targets is toadequately use the decision level r₀ of the statistic for estimatingDoppler and the r′₀ of the one for delay, respectively in (32), (37) andto enumerate several targets as functions of r₀.r′₀.

<8.1 Multiple Target Detection Using CDMT>

Consider another approach to multiple target problems. Let {(k_(d,j),

_(D,j))}^(NPath) _(j=1) be N_(path) pairs of delay and Doppler. Dividethe target space Θ′=[0,T)×[0,F) into 4 regions:

[MF544]

₁=[0,T/2)×[0,F/2),

₂=[0,T/2)×[F/2)×[F/2,F),

₃=[T/2,T)×[0,F/2),

₄=[T/2,T)×[F/2,F)

and assign TD- and FD-PCs:

[MF545]

χ={X,X′},

={Y,Y′},

={Z,Z′},

={W,W′}

to each region. Suppose that the 2-D PCs

[MF546]

χ,

,

,

have their chip address sets, defined as

[MF547]

$\begin{matrix}\left. \begin{matrix}\begin{matrix}\begin{matrix}{{\mathcal{M}_{1} = \left\{ {{\left( {m,m^{\prime}} \right){0 \leq m \leq {\frac{N}{2} - 1}}},{0 \leq m^{\prime} \leq {\frac{N^{\prime}}{2} - 1}}} \right\}},} \\{{\mathcal{M}_{2} = \left\{ {{\left( {m,m^{\prime}} \right){0 \leq m \leq {\frac{N}{2} - 1}}},{\frac{N^{\prime}}{2} \leq m^{\prime} \leq {N^{\prime} - 1}}} \right\}},}\end{matrix} \\{{\mathcal{M}_{3} = \left\{ {{\left( {m,m^{\prime}} \right){\frac{N}{2} \leq m \leq {N - 1}}},{0 \leq m^{\prime} \leq {\frac{N^{\prime}}{2} - 1}}} \right\}},}\end{matrix} \\{\mathcal{M}_{4} = {\left\{ {{\left( {m,m^{\prime}} \right){\frac{N}{2} \leq m \leq {N - 1}}},{\frac{N^{\prime}}{2} \leq m^{\prime} \leq {N^{\prime} - 1}}} \right\}.}}\end{matrix} \right\} & (80)\end{matrix}$

and their associated signature waveforms, defined by

[MF548]

$\begin{matrix}\left. \begin{matrix}{{v^{(i)}\left\lbrack {k;^{(i)}} \right\rbrack}{\sum\limits_{{({m,m^{\prime}})} \in {\mathcal{M}\; i}}{X_{m}^{(i)}X_{m}^{\prime {(i)}}_{m,M,{m^{\prime}M^{\prime}}}^{d}{{z\lbrack k\rbrack}.}}}} \\{{^{(i)} = \left( {\left\{ X_{m}^{(i)} \right\},\left\{ {X^{\prime}}_{m}^{(i)} \right\}} \right)},{X_{m}^{(i)} \in \left\{ {{- 1},1} \right\}},{\left( {m,m^{\prime}} \right) \in \mathcal{M}_{i}},}\end{matrix} \right\} & (81)\end{matrix}$

where χ⁽¹⁾, χ⁽²⁾, χ⁽³⁾, χ⁽⁴⁾ correspond to

[MF549]

χ,

,

,

.

Then the CE of the radar signal has the multiple targets form

[MF550]

$\begin{matrix}{{{\phi \lbrack k\rbrack} = {\sum\limits_{i = 1}^{4}{\sum\limits_{\overset{\_}{q}}{d_{\overset{\_}{q}} \cdot _{{q\; \; \mathcal{M}},{{qN}^{\prime}M^{\prime}}}^{1}}}}},{{\upsilon^{(i)}\left\lbrack {k;^{(i)}} \right\rbrack}.}} & (82)\end{matrix}$

If this CE is transmitted through N_(path) doubly dispersive channelswith delay, Doppler, and attenuation factor, respectively denoted by

[MF551]

{(t _(d,i) ,f _(D,i) A _(i) e ^(iπ) ^(i) )}_(i=1) ^(N) ^(path) ,

then the signal component of its received signal is given by

[MF552]

$\begin{matrix}{{r\lbrack k\rbrack} = {\sum\limits_{i = 1}^{N_{path}}{A_{i}e^{{iN}_{i}}_{t_{d,i,f_{D,i}}}^{d}{{\psi \lbrack k\rbrack}.}}}} & (83)\end{matrix}$

While, the receiver uses the 2-D PC

[MF553]

χ(or

)

for the TD-template CE of type 3 and FD-template CE of type 4

[MF554]

ψ_(ρ′,{right arrow over (ρ)}) ⁽³⁾[k] and Ψ_(ρ,{right arrow over (ρ)})⁽⁴⁾[

]

in (29), (33) and restricts the range of controlling

[MF555]

_(μ) and k _(σ),

respectively of the CCR of type 3 in (38) and of the CCR of type 4 in(43):

[MF556]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d)) and C _(ρ,{right arrow over (p)})⁽⁴⁾(k _(σ);

_(D))

to

[MF557]

${0 \leq _{\mu} \leq {\left\lfloor \frac{M^{\prime}N^{\prime}}{2} \right\rfloor - 1}},{0 \leq k_{\sigma} \leq {\left\lfloor \frac{MN}{2} \right\rfloor - 1}}$

for R₁ and to [MF558]

${\left\lfloor \frac{M^{\prime}N^{\prime}}{2} \right\rfloor \leq _{\mu} \leq {M^{\prime}N^{\prime}}},{0 \leq k_{\sigma} \leq {\left\lfloor \frac{MN}{2} \right\rfloor - 1}}$

for R₂, then it gets the MLEs for the target subspaces R₁,R₂. Similarlythe receiver gets MLEs for other regions R₃,R₄. This technique is basedon the philosophy behind the CDMA and so is referred to as thecode-division multiple target (CDMT)s.

Numerical simulations with N=N′=64, N_(path)=4, SNR ≥5 dB showed that3,4 targets are successfully detected with probability 80% by using thePUL algorithm.

<8.2 Multiple Target Detection Using Artificial Delay-Doppler>

As shown in the proof of the convergence of PUL algorithm, TD-CCF oftype 3 and FD-CCF of type 4

[MF559]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d)) and C _(ρ,{right arrow over (p)})⁽⁴⁾(k _(σ);

_(D)),

respectively using the orthogonal projection operators P₃ onto theT_(s)-time limited (TL) TD-space and P₄ onto F_(s)-band limited (BL)FD-space tell us that a pair of TFSOs of type 3 and of type 1:

[MF560]

(

,

)

(or that of the frequency dual of TFSO of type 3 and the TFSO of type 2:

[MF561]

(

,

))

is an inherent operator due to the NCP of delay and Doppler, and thusplays an essential role in radar and multiplexed communication systemsbecause such a pair contains either a pair of two unknowns

[MF562]

(

_(D) ,k _(d))

or that of two control-parameters

[MF563]

(

_(μ) ,k _(σ))

for getting a pair of two MLEs

[MF564]

(

_(D) ,{circumflex over (k)} _(d)).

On the contrary, the modulation and demodulation TFSOs

[MF565]

,

and the data-level and chip-level TFSOs

[MF566]

_(pNM,p′N′M′) ^(d)

_(ρM,0) ^(d),

_(ρ′M′,0) ^(f,d),

with data-level and chip-level addresses

[MF567]

((ρ,ρ′),{right arrow over (p)})

are independent of (

_(D),k_(d)).

While, the TD- and FD-CCF pair in (41), (45)

[MF568]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d)),C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k_(σ);

_(D)),

with address parameters

[MF569]

((ρ,ρ′),{right arrow over (p)}),

contains the data

[MF570]

d _({right arrow over (p)})

together with several PDs.

This suggests that if the artificial parameters

[MF571]

{k _(d,i),

_(D,i)}_(i=1) ^(N) ^(push)

(for simplicity setting

[MF572]

A _(i) e ^(iκ) ^(i) =1)

are embedded in

[MF573]

d _({right arrow over (p)}),

then such parameters are available for a multiple target problem. Forexample, consider communication systems based on the NCP of delay andDoppler using N_(path) pairs of TD-TFSOs and TD-CCFs and N_(path) pairsof FD-TFSOs and FD-CCFs, defined as

[MF574]

$\begin{matrix}\left. \quad\begin{matrix}{{{pair}\mspace{14mu} {of}\mspace{14mu} {TD}\text{-}{TFSOs}\text{:}\left( {_{{kd},i,_{D,i}}^{d},_{{\hat{k}}_{d,i,,p,}}^{d}} \right)},{{{TD}\text{-}{CCF}\text{:}{c_{\rho^{\prime},\overset{\_}{p}}^{(3)}\left( {_{\mu,{i;}}{\hat{k}}_{d,i}} \right)}1} \leq i \leq N_{path}},} \\{{{pair}\mspace{14mu} {of}\mspace{14mu} {FD}\text{-}{TFSOs}\text{:}\left( {_{_{D,{i - k_{d}},i}}^{f,d},_{{\hat{}}_{D,i,{- k_{d}},i}}^{f,d}} \right)},{{{FD}\text{-}{CCF}\text{:}{C_{\rho^{\prime}\overset{\_}{p}}^{(4)}\left( {k_{\sigma,{i;}}{\hat{}}_{D,i}} \right)}1} \leq i \leq {N_{path}.}}}\end{matrix} \right\} & (84)\end{matrix}$

with N_(path) pairs of parameters

[MF575]

{(k _(d,i),

_(D,i))}_(i=1) ^(N) ^(path) .

Such systems use N_(path) pairs of TD-, FD-templates embedded intransmitted TD- and FD-signatures and PUL algorithm for estimating

[MF576]

(k _(d,i),

_(D,i)),

where the PUL algorithm was called the “active” PUL[25, 30] in the sensethat updating is done at the transmitter; but the adjective active wassomething of a misnomer because updating is applicable to the receiveronly. The above MLE is based on identifying N_(path) pairs of knownshifts

[MF577]

{(k _(d,i),

_(D,i))}_(i=1) ^(N) ^(path)

from signals with embedded those shifts. Namely, it is relativelydifferent from the single-target problem. Hence the above PUL is anordinary PUL.

The CDMT technique can be applied to

[MF578]

-ary

phase-shift-keying (PSK) communication with data

[MF579]

${d_{\overset{\_}{q}} = W_{\mathcal{M}}^{- {kq}}},{W_{\mathcal{M}} = e^{{- i}\frac{2\pi}{\mathcal{M}}}},{k_{\overset{\_}{q}} \in {\mathbb{Z}}},{0 \leq k_{\overset{\_}{q}} \leq {\mathcal{M} - 1.}}$

Using N_(path) 2-D PCs and two twiddle factors

[MF580]

,

, 0≤

,

≤

−1

placed in the front of N′ TD-CCF and N FD-CCF arrays

[MF581]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(μ) ;{circumflex over (k)} _(d)) and C _(ρ,{right arrow over (p)})⁽⁴⁾(k _(σ);

_(D)),

called a “phase tuned layer (PTL)” and replacing the

[MF582]

$\underset{\rho^{\prime}_{\mu}}{\arg \mspace{11mu} \max},\underset{\rho,k_{\sigma}}{\arg \mspace{14mu} \max}$

operations for MLE in the PUL by

[MF583]

$\underset{{\rho^{\prime}_{u}},}{\arg \mspace{11mu} \max};\underset{\rho,k_{\sigma},^{\prime}}{\arg \mspace{14mu} \max};$

by augmenting two PTL variables

[MF584]

,

.

one can get a PTL that plays a role of replacing

[MF585]

d _({right arrow over (p)})

of the pair of CCFs in (41), (45) by

[MF586]

d _({right arrow over (p)}) ·

,d _({right arrow over (p)})·

with the result that phase errors in the real parts of the CCFs haveGaussian distribution with mean 0.

The inventor can get good numerical simulation results with N=N′=16,

[MF587]

=8,16

and SNR 30 dB. However, when

[MF588]

=16,

sidelobes arise the left- and right-hand sides of the main lobe in thedistribution, which means decoding errors may occur. Hence only use ofsimple PD-cancellation methods limit to setting

[MF589]

=8.

Thus, in order to try M-ary communication with

[MF590]

≥32,

one can divide the delay-Doppler space Θ′ and the signal TFP S into

[MF591]

$\frac{\mathcal{M}}{8}$

delay-Doppler sub-spaces and sub-TFPs and assign

[MF592]

$\left\{ {{X^{(i)} = \left( {\left\{ X_{m}^{(i)} \right\},\left\{ X_{m}^{\prime {(i)}} \right\}} \right)},X_{m}^{(i)},{X_{m}^{\prime {(i)}} \in \left\{ {{- 1},1} \right\}}} \right\}_{i = 1}^{\frac{\mathcal{M}}{8}}$

to each sub-TFP. Moreover, let us suppose that artificial shifts

[MF593]

(k _(d,i),

_(D,i))

are around the center of i-th delay-Doppler sub-space and use the PTLsof W₈=exp(−i2π/8) and W_(M)=exp(−i2π/M), placed in the front of TD- andFD-CCFs, namely,

[MF594]

$\begin{matrix}\begin{matrix}{{W_{8}^{\lambda \overset{\_}{p}} \cdot W_{\mathcal{M}}^{j\; \tau}},{{for}\mspace{14mu} {TD}\text{-}{CCF}},{\lambda_{\overset{\_}{p}} \in \left\{ {0,1,\ldots \mspace{14mu},7} \right\}},{j_{\tau} \in \left\{ {0,1,\ldots \mspace{14mu},\frac{\mathcal{M}}{8},} \right\}}} \\{{W_{8}^{\lambda_{\overset{\_}{p}}^{\prime}} \cdot W_{\mathcal{M}}^{j\; F}},{{for}\mspace{14mu} {FD}\text{-}{CCF}},{\lambda_{\overset{\_}{p}}^{\prime} \in \left\{ {0,1,\ldots \mspace{14mu},7} \right\}},{j_{F} \in \left\{ {0,1,\ldots \mspace{14mu},{\frac{\mathcal{M}}{8}.}} \right\}}}\end{matrix} & (85)\end{matrix}$

This results in the following replacement of the PTL for

[MF595]

d _({right arrow over (p)})

and arg max operation

[MF596]

$\begin{matrix}\left. \begin{matrix}{\left. d_{\overset{\_}{p}}\Rightarrow{d_{\overset{\_}{p}} \cdot W_{8}^{\lambda \overset{\_}{p}} \cdot W_{\mathcal{M}}^{j\; \tau}} \right.,{\arg \mspace{14mu} \max_{\rho^{\prime},{_{p};\lambda_{\overset{\_}{p}};{j\; \tau}}}},} & {{{for}\mspace{14mu} {TD}\text{-}{CCF}},} \\{\left. d_{\overset{\_}{p}}\Rightarrow{d_{\overset{\_}{p}} \cdot W_{8}^{\lambda \overset{\_}{p}} \cdot W_{\mathcal{M}}^{j\; F}} \right.,{\arg \mspace{14mu} \max_{\rho,{k_{\sigma};\lambda_{\overset{\_}{p}}^{\prime}},{j\; F},}}} & {{for}\mspace{14mu} {FD}\text{-}{{CCF}.}}\end{matrix} \right\} & (86)\end{matrix}$

Hence one should require

[MF597]

$\frac{\mathcal{M}}{8} \cdot 8 \cdot \left( {N + N^{\prime}} \right)$

CCFs in total. Therefore, one can obtain numerical decoding simulationresults for 128-PSK, 256-PSK using 16, 32 2-D PCs. This M-ary PSKcommunication realized a new class of multiplexing communication, called“delay-Doppler space division multiplex (dD-SDM)”[35, 34]. However, thephase resolution of this system remains within W_(M)=exp(−i2π/M). So, itis consequently unable to realize a high

[MF598]

-ary PSK communication. A new technique to realize a high M-ary PSKcommunication is proposed as follows.The inventor can give communication systems with modulation anddemodulation of high

[MF599]

-PSK in cooperating with establishing synchronisation, that are capableof being used also as a synchroniser for communication and or radar.First of all, one can provide explanations of FIGS. 12-17 relating to

[MF600]

-ary PSK encoder and decoder as follows.

FIG. 12 illustrates the block-diagram of a transmitter (or an encoder)that is capable of being used also as an efficient and high-resolutionradar equipped with

[MF601]

-ary PSK communication.

There are two different kinds of states: one is synchronizer/radar state(when the data d_(q→) is on the upward sides of the switches 1-1,1-2)and another is

[MF602]

-ary PSK communication state (when the data d_(q→) is connected to thedown-ward of the switches 1-1,1-2). Each state is controlled by theswitches. When

[MF603]

d _({right arrow over (q)})

is

[MF604]

-ary PSK, i.e., the data communication state, the switches are ondownward.

The leftmost part of the transmmitter is the input

[MF605]

d _({right arrow over (q)}).

When the transmitter is in the synchronizer/radar state, it performsfour procedures to a chip waveform:

[MF606]

$\begin{matrix}{\begin{pmatrix}\left( {m^{\prime},\overset{\_}{q}} \right) & {\text{-}{TD}\text{-}{template}} \\\left( {m,\overset{\_}{q}} \right) & {\text{-}{FD}\text{-}{template}}\end{pmatrix}\mspace{14mu} {generation}\mspace{14mu} {by}\mspace{14mu} {the}\mspace{14mu} 2\text{-}D\mspace{11mu} {PC}_{}} \\{\&\left( {N^{\prime}/N} \right)\text{-}{TD}{\text{-}/{FD}}\text{-}{template}\mspace{14mu} {multiplexing}} \\{{\&\left( {P^{\prime}/P} \right)\text{-}{TD}{\text{-}/{FD}}\text{-}{signature}\text{-}},{{and}\mspace{14mu} d_{\overset{\_}{q}}\text{-}{multiplexing}}}\end{matrix}$

while when the transmitter is in the data communication state, itperforms six procedures to a chip waveform:

[MF607]

$\begin{matrix}{\begin{pmatrix}\left( {m^{\prime},\overset{\_}{q}} \right) & {\text{-}{TD}\text{-}{template}} \\\left( {m,\overset{\_}{q}} \right) & {\text{-}{FD}\text{-}{template}}\end{pmatrix}\mspace{14mu} {generation}\mspace{14mu} {by}\mspace{14mu} {the}\mspace{14mu} 2\text{-}D\mspace{11mu} {PCs}_{}^{(i)}} \\{\&\left( {N_{1}^{\prime}/N_{1}} \right)\text{-}{TD}{\text{-}/{FD}}\text{-}{template}\mspace{14mu} {multiplexing}} \\{\&\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)\text{-}{PC}\mspace{14mu} {multiplexed}\mspace{14mu} {signature}} \\{\&\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)\text{-}{signature}\mspace{14mu} {multiplexing}} \\{{\&\left( {P^{\prime}/P} \right)\text{-}{TD}{\text{-}/{FD}}\text{-}{signature}\text{-}},{{and}\mspace{14mu} d_{\overset{\_}{q}}\text{-}{multiplexing}}}\end{matrix}$

Next, in the block of the k-encoder, consisted of the j′-th AC chosenfrom

[MF608]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

parallel ACs, where the symbol

[MF609]

denotes the set of the ACs and the dot . simply its one.

In accordance with an embodiment of the present invention, theencoder-switches 1-3, 1-4 choose the j-th AC in the block of thek-encoder, according to j, and the transmitter phase-modulates theoutput of the encoder by the j′-th PSK symbol

[MF610]

which is determined by an encoding of k with integers j,j′, defined as

[MF611]

${k = {{j\; \mathcal{M}_{0}} + j^{\prime}}},{j = \left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack},{j^{\prime} = {\left\{ \frac{k}{\mathcal{M}_{0}} \right\}.}}$

When the switch 1-2 is on upward and downward sides, respectively

[MF612]

TD-CEψ[k] and FD-CEΨ[

]

in (27) and

[MF613]

TD-CEψ^(AC)[k] and FD-CEΨ^(AC)[

]

as defined below in (87) respectively, might be passed through theswitch. Furthermore, in accordance with an embodiment of the presentinvention, the transmitter (or the encoder) modulates those CEs by acarrier

[MF614]

_(c),

transmits the resultant passband (PB) signal through the main channel(MC) with shifts

[MF615]

(k _(d),

_(D)),

and demodulates the noisy PB signal, contaminated by noise, by thecarrier

[MF616]

_(c),

resulting a received signal.

The explanation of the block-diagram shown in FIG. 12 is over.

If a high

[MF617]

-PSK modulated symbol

[MF618]

$\exp \left( \frac{i\; 2\; \pi \; k}{\mathcal{M}} \right)$

is transmitted and is contaminated by both phase noise and additivenoise, then a signal

[MF619]

$\exp \left( \frac{i\; 2\; \pi \; k}{\mathcal{M}} \right)$

should be resolved. However, its resolution is unrealizable. To solveit, in a k-encoder (the right block in the lower part of FIG. 12), thetransmitter encodes k in the form

[MF620]

${k = {{j\; \mathcal{M}_{0}} + j^{\prime}}},{j = \left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack},{j^{\prime} = \left\{ \frac{k}{\mathcal{M}_{0}} \right\}}$

so as to transmit a lower

[MF621]

₀

-PSK modulated symbol

[MF622]

${\exp \left( \frac{i\; 2\; \pi \; j^{\prime}}{\mathcal{M}_{0}} \right)}.$

While to transmit the encoded integer

[MF623]

${j = \left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack},$

the transmitter relies heavily on the joint estimation method of delayand Doppler with high precision, proposed by this disclosure. Namely,the transmitter firstly divides the (t_(d), f_(D))-parameter space Θ′and the signal time-frequency plane (TFP) S equally into

[MF624]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

sub-parameter spaces Θ^((i),′) and sub-TFP S^((i)); secondly assigns 2-DPC χ^((i)) to each sub-TFPs; thirdly 2-DBPSK modulates a chip waveformby

[MF625]

$\left\{ \chi^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}$

2-D PCs; fourthly combines those BPSK-modulated signals by

[MF626]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right) - \chi^{(i)}$

-multiplexing; fifthly assumes a situation that the resultantmultiplexed signal is transmitted through the j-th channel, called thej-th artificial channel (AC), chosen from

[MF627]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

parallel ACs, according to j, so that those multiplexed signal istime-frequency shifted by about the center of the sub-parameter spaceΘ^((j),′); sixthly PSK-modulates the output of the j-th AC by the j′-th

[MF628]

₀

-PSK symbol

[MF629]

$W_{\mathcal{M}_{0}}^{- j^{\prime}} = {\exp \left( \frac{i\; 2\; \pi \; j^{\prime}}{\mathcal{M}_{0}} \right)}$

to get a transmit CE. The transmitter modulates this CE by the carrier

[MF630]

_(c)

and inputs the resultant PB signal to the MC with shifts

[MF631]

(k _(d),

_(D)).

External noise is added to the output of the MC. The transmitterdemodulates the noisy PB signal by the carrier

[MF632]

_(c).

The resultant BB signal yields a received CE.

Next, the explanation of the block-diagram as shown in FIG. 13 is given.

FIG. 13 shows a block-diagram of a receiver/synchroniser (or a decoder)of a communication system of capable of being used also as an efficientand high-resolution radar, equipped with

[MF633]

-ary PSK communication.

When the receiver is in the synchronizer/radar state (i.e., the switch2-1 is on the upward side), it performs two procedures to a chipwaveform:

[MF634]

$\begin{pmatrix}{\left( {\rho^{\prime},\overset{\rightarrow}{q}} \right)\mspace{11mu} \text{-}{TD}\text{-}{template}} \\{\left( {\rho,\overset{\rightarrow}{q}} \right)\mspace{11mu} \text{-}{FD}\text{-}{template}}\end{pmatrix}\mspace{14mu} {generation}\mspace{14mu} {by}\mspace{14mu} {the}\mspace{14mu} 2\text{-}D\mspace{11mu} {PC}\; \chi$

while when the receiver is in the data communication state (i.e., theswitch 2-1 is on the downward side), it firstly decodes k in thek-decoder, denoted by k{circumflex over ( )} and defined as

[MF635]

${\hat{k} = {{\hat{j}\; \mathcal{M}_{0}} + \hat{j^{\prime}}}},{\hat{j} = \left\lbrack \frac{\hat{k}}{\mathcal{M}_{0}} \right\rbrack},{\hat{j^{\prime}} = \left\{ \frac{\hat{k}}{\mathcal{M}_{0}} \right\}},$

and secondly performs two procedures a chip waveform:

[MF636]

$\quad{\begin{pmatrix}{\left( {\rho^{\prime},\overset{\rightarrow}{q}} \right)\mspace{20mu} \text{-}{TD}\text{-}{template}} \\{\left( {\rho,\overset{\rightarrow}{q}} \right)\mspace{20mu} \text{-}{FD}\text{-}{template}}\end{pmatrix}\mspace{14mu} {generation}\mspace{14mu} {by}\mspace{14mu} {the}\mspace{14mu} 2\text{-}D\mspace{14mu} {PCs}\; \chi^{(\hat{j})}}$

so as to select the j{circumflex over ( )}-th AC from

[MF637]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

parallel ACs, where the symbol

[MF638]

denotes the set of the ACs and the dot . simply its one.

In accordance with an embodiment of the present invention, when theswitch 2-1 is on downward side, the decoder chooses the

[MF639]

ĵ

-th AC in the block of the k-decoder, according to j{circumflex over( )}, and the receiver phase-modulates the output of the AC by thej^({circumflex over ( )})′-th PSK symbol

[MF640]

to get a template CE. The receiver calculates the CCR between thereceived CE and this template CE, in the COR-block of the lower part ofthe diagram of FIG. 13.

The symbol COR means correlation. The big block in the right-hand-sideof the COR indicates the block of maximum likeklihood estimating

[MF641]

(k _(d),

_(D)),

and Ae^(iκ), recovering

[MF642]

d _({right arrow over (q)}),

and decoding k{circumflex over ( )} for the selected chip-leveladdresses (ρ′,ρ): When the receiver is in the synchroniser/radar state(i.e., the switch 2-2 is on the upward side), it performs

[MF643]

$\mspace{20mu} \left. {\underset{\rho^{\prime},_{\mu}}{argmax}\; \left. {\left\lbrack {{c_{\rho^{\prime},\overset{\rightarrow}{p}}^{(3)}\left( {_{\mu};{\hat{k}}_{d}} \right)}\text{/}\left( {X_{\rho^{\prime}}^{\prime},{A\; e^{i\; \kappa}}} \right)} \right\rbrack}\Updownarrow{PUL} \right.\mspace{14mu} k_{\sigma}^{*}}\rightarrow\left. {{\hat{k}}_{d}\mspace{11mu} _{\mu}^{*}}\rightarrow\left. {\hat{}}_{D}\rightarrow\left. \begin{pmatrix}{{MLE}\text{:}\left( {k_{\sigma}^{*},_{\mu}^{*}} \right)} \\{{MLE}\text{:}\; \hat{A}\; e^{i\; \hat{\kappa}}}\end{pmatrix}\rightarrow{d_{ij}\mspace{14mu} {reconstruction}} \right. \right. \right. \right.$$\mspace{20mu} {\underset{\rho,k_{\sigma}}{argmax}\; {\left\lbrack {{C_{\rho^{\prime},\overset{\rightarrow}{p}}^{(4)}\left( {k_{\sigma};{\hat{}}_{D}} \right)}\text{/}\left( {X_{\rho},{A\; e^{i\; \kappa}}} \right)} \right\rbrack}}$

While when the receiver is in the data communication state (i.e., theswitch 2-2 is on the downward side), it performs

[MF644]

$\mspace{20mu} \left. {\underset{\rho^{\prime},_{\rho},\overset{\sim}{j},\hat{j^{\prime}}}{argmax}\; \left. {\left\lbrack {{c_{\rho^{\prime},\overset{\rightarrow}{p},\hat{j},\hat{j^{\prime}}}^{{A\; C},{(3)}}\left( {{_{\mu};{\hat{k}}_{d}},j,j^{\prime}} \right)}\text{/}\left( {X_{p^{\prime}}^{\prime}A\; e^{i\; \kappa}} \right)} \right\rbrack}\Updownarrow{PUL} \right.\mspace{14mu} k_{\sigma}^{*}}\rightarrow\left. {{\hat{k}}_{d}\mspace{14mu} _{\mu}^{*}}\rightarrow\left. {\hat{}}_{D}\rightarrow\left. \begin{pmatrix}{{MLE}:\left( {k_{\sigma}^{*},_{\mu}^{*}} \right)} \\{{MLE}:{\hat{A}\; e^{i\; \hat{\kappa}}}}\end{pmatrix}\rightarrow\begin{matrix}{k\mspace{14mu} {decoding}} \\{\hat{k} = {{j^{*}\mathcal{M}_{0}} + j^{\prime,*}}}\end{matrix} \right. \right. \right. \right.$$\mspace{20mu} {\underset{\rho,k_{\sigma},\hat{j},{\hat{j}}^{\prime}}{argmax}\; {\left\lbrack {{C_{\rho^{\prime},\overset{\rightarrow}{p},\hat{j},\hat{j^{\prime}}}^{{A\; C},{(4)}}\left( {{k_{\sigma};{\hat{}}_{D}},j,j^{\prime}} \right)}\text{/}\left( {X_{\rho}A\; e^{i\; \kappa}} \right)} \right\rbrack}}$

The explanation of the block-diagram shown in FIG. 13 is over.

Next, consider a situation that a received CE is 2-DBPSK demodulated bya 2-D PC phase modulated chip waveform. When the receiver is in the thedata communication state (i.e., the switch 2-1 is on downward), it 2-DBPSK modulates a chip waveform by the estimated

[MF645]

ĵ

-th 2-D PC [MF646]

χ^((ĵ))

and inputs the 2-DBPSK modulated signal to the

[MF647]

ĵ

-th AC, chosen from

[MF648]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

parallel ACs and

[MF649]

₀

-PSK demodulates the output signal of the AC by the symbol

[MF650]

.

In the final step in the MLE block, the real parts of TD-CCFs andFD-CCFs, normalized by the PC and the attenuation factor

[MF651]

[c _(ρ′,{right arrow over (p)},ĵ,ĵ′) ^(AC,(3))(

_(μ) ;{circumflex over (k)} _(d) ,j,j′)/(X′ _(ρ′) Ae ^(iκ))],

[C _(ρ′,{right arrow over (p)},ĵ,ĵ′) ^(AC,(4)),(k _(σ);

_(D) ,j,j′)/(X _(ρ) Ae ^(iκ))]

are maximized in terms of 4 variables

[MF652]

(ρ′,{right arrow over (p)},ĵ,ĵ′),(ρ′,{right arrow over (p)},ĵ,ĵ′)

to get MLEs of [MF653]

(k _(d),

_(D)).

If the PUL algorithm converges, then the receiver gets convergence

[MF654]

(k* _(d),

*_(D)).

Thus the convergence is common in two states. When the receiver is inthe synchroniser/radar state (i.e., the switch 2-2 is on the upwardside), it recovers

[MF655]

d _({right arrow over (q)}),

while, when the receiver is in the data communication state (i.e., theswitch 2-2 is on the downward side), it decodes k in the form

[MF656]

k*=j*

₀ +j* ^(,′)

by using MLEs for

[MF657]

(ĵ,ĵ′).

Note that even when the receiver is in the usual synchroniser/radarstate (i.e., the switch 2-2 is on the upward side), it is capable of low

[MF658]

₀

-PSK modulating/demodulating to convey data

[MF659]

d _({right arrow over (q)}).

e.g.,

[MF660]

₀=8.

FIG. 14 shows the distribution of magnitudes of the real parts of theCCFs on the main channel (MC)'s delay τ and Doppler τ parameter space.It indicates that the main lobe is isolated.

When the transmitter is in the synchroniser/radar state, the main lobelocates at

[MF661]

(k _(d),

_(D))

and it is discriminated from side lobes, where the double-wave symboldenotes background noise.

FIG. 15 shows the distribution of magnitudes of the real parts of theCCFs on the τ-ν plane when three parallel ACs are added to the MC.

Three parallel ACs are connected in series in the front of the MC inorder to encode k using the

[MF662]

-PSK modulated signal. The MC still remains at

[MF663]

(k _(d),

_(D)).

However, when the ACs are added to the MC and the three series channels,MC+AC0,MC+AC1,MC+AC2 locate at

[MF664]

(k _(d) +k _(d) ⁽⁰⁾,

_(D)+

_(D) ⁽⁰⁾),(k _(d) +k _(d) ⁽¹⁾,

_(D)+

_(D) ⁽¹⁾),(k _(d) +k _(d) ⁽²⁾,

_(D)+

_(D) ⁽²⁾),

respectively.The time shifts and frequency shifts are simply additive in quantitativeterms, However, several new PDs simultaneously arise and are accompaniedby group-theoretical property. In order to obtain the distribution asshown in FIG. 15, one needs the rigorous estimation of several PDs andmaximization of the real part of the CCFs by using MLE, together withaugmenting parameter variables j,j′ to cancel out the PDs, as describedin written description of this disclosure. The augmented variables j,j′may take

[MF665]

${\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1},{\mathcal{M}_{0}.}$

different values, respectively.

FIG. 16 shows the symbol's fundamental TFP S of [0, T_(s))×[0, F_(s))and its 4 partitioned sub-TFPs S^((i)). This partitioning is intended toexploit the AC parameter space of co-dimension 2 having the NCP of TFSs,where the term “co-dimension” means the dimension of the parameterspace.

In FIG. 16, the 4-partition (Gabor's partition) of the signal's TFP oftime duration T_(s) and bandwidth F_(s) (S⁽⁰⁾, S⁽¹⁾, S⁽²⁾, S⁽³⁾) and thevertical axis is perpendicularly added to the TFP, where a scale of 4pairs of AC shifts

[MF666]

(k _(d) ⁽⁰⁾ ,

D ⁽⁰⁾),(k _(d) ⁽¹⁾

_(D) ⁽¹⁾),(k _(d) ⁽²⁾,

_(D) ⁽²⁾),(k _(d) ⁽³⁾,

_(D) ⁽³⁾),

is attached to it, associated as a third variable of the TFP besides theT and F coordinate axes.

FIG. 17 shows also the 4-partitioned sub-TFPs S^((i)) with assigned PCX^((i)) that is intended to exploit the AC parameter space ofco-dimension 2 having the NCP of TFSs.

As shown in FIG. 17, each S^((i)) is time-frequency shifted by (k^((i))_(d),l^((i)) _(D), 0≤i≤3. The signal's TFP S can be identified with thetarget space of the parameter of the MC

[MF667]

(k _(d),

_(D))

having its fundamental unit plane

[MF668]

[0,T _(s))×[0,F _(s)).

That is, if a target exists around the TFP with data-level address

[MF669]

{right arrow over (p)}=(p,p′)

i.e., around the neighborhood of sub-TFP[(p−1)T_(s)·pT_(s))×[(p′−1)F_(s),p′F_(s)) then the target should bediscriminated within the data-level address

[MF670]

{right arrow over (p)}=(p,p′), 1≤p≤P−1, 1≤p′≤P′−1

Hence the argmax operation on the two real parts of the CCFs in FIG. 13needs the data-level address

[MF671]

{right arrow over (p)}

regardless of the use of data communication.

As discussed above, in the transmitter and receiver, respectively inFIGS. 12 and 13 the upward and downward systems are alternativelyswitched.

The systems connected to the upward sides of the switches 1-1, 1-2, 2-1,2-2 in FIGS. 12 and 13 are those in accordance with an embodiment of thepatent[6], while in an embodiment of the present disclosure, the systemsconnected to the downward sides of these switches are multiplexedcommunication systems based on the encoder/decoder exploiting the NCP ofthe TFSs.

FIGS. 12 and 13 enables us to discriminate against thesynchroniser/radar not equipped with high M-ary communication and toshow originality in this disclosure.

In accordance with an embodiment of this disclosure, in the transmitterand receiver system, in order to transmit

[MF672]

-ary PSK

modulated symbol

[MF673]

${\exp \left( \frac{i\; 2\; \pi \; k}{\mathcal{M}} \right)},{0 \leq k \leq {\mathcal{M} - 1}}$

and to embed efficiently “information” k into its transmit signal, foran integer

[MF674]

₀, 1≤

₀≤

,

the transmitter prepares

[MF675]

$\left( {{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack \left( {\overset{def}{=}{{the}\mspace{14mu} {integer}\mspace{14mu} {part}\mspace{14mu} {of}\mspace{14mu} \frac{\mathcal{M} - 1}{\mathcal{M}_{0}}}} \right)} + 1} \right)$

pairs of 2-DBPSK codes, i.e., pairs of TD-PCs of period N

[MF676]

X ^((j)) ={X _(m) ^((j))}

and FD-PCs of period N′

[MF677]

X ^((j),) ′={X _(m) ^((j),t)}

and divides the fundamental unit plane Θ′=[0,T_(s))×[−F_(s)/2, F_(s)/2)of the delay and Doppler-parameter space

[MF678]

Θ_(max)=[0,T _(max))×[−F _(max)/2,F _(max)/2)

and the signal's TFP S=[0,T_(s))×[0,F_(s)) equally into

[MF679]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

sub-parameter spaces and sub-TFPs, respectively denoted as

[MF680]

$\Theta^{(i)},^{(i)},{0 \leq i \leq \left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack},$

where T_(s)=NMΔt, F_(s)=N′M′Δf denote the time duration and band-widthof the symbol

[MF681]

d _({right arrow over (q)}) ,{right arrow over (q)}=(q,q′),

T_(max)=PT_(s), F_(max)=P′F_(s) denote the maxima of delay and Dopplerto be detected with integers P, P′, and the sampling intervals Δt, Δf.Next the transmitter decomposes information k given as

[MF682]

$k = {{{\left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack \mathcal{M}_{0}} + \left\{ \frac{k}{\mathcal{M}_{0}} \right\}} = {{j\; \mathcal{M}_{0}} + j^{\prime}}}$

and transmits an integer pair (j, j′), i.e., an integer part

[MF683]

${j = {\left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack \left( {\overset{def}{=}{{the}\mspace{14mu} {integer}\mspace{14mu} {part}\mspace{14mu} {of}\mspace{11mu} \frac{k}{\mathcal{M}_{0}}}} \right)}},{0 \leq j \leq \left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack}$

and its fraction fraction part

[MF684]

${j^{\prime} = {\left\{ \frac{k}{\mathcal{M}_{0}} \right\} \left( {\overset{def}{=}{{the}\mspace{14mu} {fractional}\mspace{14mu} {part}\mspace{14mu} {of}\mspace{14mu} \frac{k}{\mathcal{M}_{0}}}} \right)}},{0 \leq j^{\prime} \leq {\mathcal{M}_{0} - 1}},$

in place of k. To transmit the encoded integer pair (j, j′), thetransmitter 1) 2-D BPSK-modulates a chip waveform of time durationT_(c)=MΔt and bandwidth F_(c)=MΔf by 2-D PC X^((j)); 2) gets a 2-Dphase-modulated signal, called a TD-signature and its Fourier Transform(FT), called an FD-signature; 3) time-frequency shifts by data-levelshift (T_(s), F_(s)) and combines these time-frequency shiftedsignatures in a non-overlapping form to obtain a

[MF685]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)\mspace{14mu} \text{-}{signature}\text{-}{multiplexed}$

signal. Next the transmitter assumes that the multiplexed signal maytransmit through an artificial channel (AC) with delay-Doppler shifts

[MF686]

(k _(d) ^((j)),

_(D) ^((j))),

located at around the center of the sub-parameter space Θ^((j),′). Tosimulate such a situation, the transmitter makes two TFSOs of type 4

[MF687]

,

act on the multiplexed TD- and FD-signals, respectively. Thirdly, thetransmitter PSK-modulates the resulting time-frequency shifted TD-signalby

[MF688]

₀-ary

symbol

[MF689]

[MF690]

$\left( {W_{\mathcal{M}_{0}} = {\exp \left( \frac{{- i}\; 2\; \pi}{\mathcal{M}_{0}} \right)}} \right),$

(i.e., multiplies the PSK-modulated signal by the symbol

[MF691]

d _({right arrow over (q)}),

), and combines these (P/P′) signals in non-overlapping form on the TFPto generate TD-CE and its DFT, FD-CE (cf. FIG. 12), equipped with

[MF692]

₀

-ary PSK communication, in place of (27), given by

[MF693]

$\begin{matrix}\left. \begin{matrix}{{{\psi^{A\; C}\left\lbrack {{k;\left\{ \chi^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}},j,j^{\prime}} \right\rbrack} = {\frac{1}{\sqrt{{PP}^{\prime}}}{\sum\limits_{q,{q^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}{\frac{d_{\overset{\rightarrow}{q}}^{\prime} \cdot W_{\mathcal{M}_{0}}^{- i^{\prime}}}{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1}_{k_{d}^{(j)},_{D}^{(j)}}^{d}{\sum\limits_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}_{{qNM},{q^{\prime}N^{\prime}M^{\prime}}}^{d}}}}}},{v\left\lbrack {k;\chi^{(i)}} \right\rbrack},} \\{{{\Psi^{A\; C}\left\lbrack {{;\left\{ \chi^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}},j,j^{\prime}} \right\rbrack} = {\frac{1}{\sqrt{{PP}^{\prime}}}{\sum\limits_{q,{q^{\prime} = 0}}^{{P - 1},{P^{\prime} - 1}}{\frac{d_{\overset{\rightarrow}{q}}^{\prime} \cdot W_{\mathcal{M}_{0}}^{- j^{\prime}}}{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1}_{_{D}^{(j)},{- k_{d}^{(j)}}}^{f,d}{\sum\limits_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}_{{q^{\prime}N^{\prime}M^{\prime}},{- {qNM}}}^{f,d}}}}}},{V\left\lbrack {;\chi^{(i)}} \right\rbrack},}\end{matrix} \right\} & (87)\end{matrix}$

where

[MF694]

υ[k;χ ^((i))],V[

;χ^((i))]

denote TD- and FD-signatures (cf. (25)). defined by

[MF695]

$\begin{matrix}\left. \begin{matrix}{{{v\left\lbrack {k;\chi^{(i)}} \right\rbrack} = {\frac{1}{\sqrt{N_{1}N_{1}^{\prime}}}{\sum\limits_{{({m,m^{\prime}})} \in I^{(i)}}{X_{m}^{(i)}X_{m^{\prime}}^{{(i)},\prime}_{{mM},{m^{\prime}M^{\prime}}}^{d}{g\lbrack k\rbrack}}}}},} \\{{{V\left\lbrack {;\chi^{(i)}} \right\rbrack} = {\frac{1}{\sqrt{N_{1}N_{1}^{\prime}}}{\sum\limits_{{({m,m^{\prime}})} \in I^{(i)}}{X_{m}^{(i)}X_{m^{\prime}}^{{(i)},\prime}_{{m^{\prime}M^{\prime}},{- {mM}}}^{f,d}{G\lbrack \rbrack}}}}},}\end{matrix} \right\} & (88)\end{matrix}$

where

[MF696]

$I^{(i)},{0 \leq i \leq \left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack}$

denotes the i-th sub-set of the chip address set {(m,m′)}^(N−1,N′−1)_(m=0,m′=0) of S, associated with the i-th sub-TFP S^((i)) as shown inFIGS. 16 and 17 and

[MF697]

${N_{1} = \frac{N}{\sqrt{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1}}},{N_{1}^{\prime} = {\frac{N^{\prime}}{\sqrt{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1}}.}}$

The TD- and FD-CEs in (87) are designed to convey the encoded integers(j,j′) of k to receivers through the MC. However, M-ary communicationleads us to set the periods of the TD-PCs and FD-PCs to be nearly

[MF698]

$\sqrt{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1}$

multiples of those for synchronization of communication or targetdetection to keep estimation precision (a situation that the switches(1-1,1-2,2-1,2-2) are on the downward and upward sides in FIGS. 12 and13 corresponds to with/without the use of data communication).

Eq. (87) tells us that TD- and FD-signatures

[MF699]

υ[k;χ ^((i))],V[

;χ^((i))]

are firstly 2-D BPSK-modulated by 2-D PC X^((i)) and secondly theresultant phase modulated signatures are combined in non-overlapped formsuch as a

[MF700]

${{\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right) - ^{(i)}}\&}\mspace{14mu} \text{-}{signature}\text{-}{multiplexed}$

signal and thirdly such a doubly-multiplexed signal is PSK-modulated by

[MF701]

₀-ary

symbol

[MF702]

and fourthly the PSK-modulated signal is transmitted through the j-th ACwith shifts

[MF703]

(k _(d) ^((j)),

_(D) ^((j))),

and fifthly is PSK-modulated by

[MF704]

₀-ary

data symbol

[MF705]

d′ _({right arrow over (q)})·

The final signal is the TD-CE

[MF706]

$\psi^{AC}\left\lbrack {{k;\left\{ ^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}},j,j^{\prime}} \right\rbrack$

and its DFT, i.e., the FD-CE [MF707]

$\psi^{AC}\left\lbrack {{;\left\{ ^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}},j,j^{\prime}} \right\rbrack$

where

[MF708]

d′ _({right arrow over (q)})

with data-level address q^(→)=(q, q′) has the value of 1.

At the receiver, an estimated k, denoted by k{circumflex over ( )}

[MF709]

{circumflex over (k)}, 0≤{circumflex over (k)}≤

−1

is decomposed with a decoded pair of integers

[MF710]

${\hat{j} = \left\lbrack \frac{\hat{k}}{\mathcal{M}_{0}} \right\rbrack},{0 \leq \hat{j} \leq \left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack},{{\hat{j}}^{\prime} = {{\left\{ \frac{\hat{k}}{\mathcal{M}_{0}} \right\} 0} \leq {\hat{j}}^{\prime} \leq {\mathcal{M}_{0} - 1.}}}$

The receiver first makes the estimated TD- and FD-TFSOs of type 4

[MF711]

and

representing the j{circumflex over ( )}-th-AC with shifts

[MF712]

(k _(d) ^((ĵ)),

_(D) ^((ĵ)))

act on such TD- and FD-signatures, respectively and secondlyPSK-demodulates those time-frequency shifted TD- and FD-signatures by

[MF713]

₀-ary

to cancel out the PSK-modulation phase (cf. FIG. 13).The CCFs of type 3 and of type 4 exploiting AC shifts, with encodedinteger pair (j,j′) and its augmented pair for argmax operations

[MF714]

(ĵ,ĵ′),

are concretely given by respectively, in place of (38) and (43)

[MF715]

$\begin{matrix}{{{c_{p^{\prime},\overset{\sim}{p},\hat{j},{\hat{j}}^{\prime}}^{{AC},{(3)}}\left( {{_{\mu};{\hat{k}}_{d}},j,j^{\prime}} \right)} = {{Ae}^{i\; \kappa}W^{k_{d}}_{c}{\sum\limits_{k \in {\mathbb{Z}}}{_{k_{d},_{D}}^{d}{\psi^{AC}\left\lbrack {{k;\left\{ ^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}},j,j^{\prime}} \right\rbrack} \times \left( {W^{\hat{k_{d}}_{c}}_{{\hat{k}}_{d}_{c}}^{d}W_{\mathcal{M}_{0}}^{- {\hat{j}}^{\prime}}_{k_{d}^{(j)}_{D}^{(j)}}^{d}_{{p\; {NM}},{p^{\prime}N^{\prime}M^{\prime}}}^{d}\frac{X_{\rho^{\prime}}^{{(j)}^{\prime}}}{\sqrt{N_{1}}}{\sum\limits_{n \in I_{T}^{(j)}}{_{n,M,{p^{\prime}M^{\prime}}}^{d}X_{n}^{(j)}{g\lbrack k\rbrack}}}} \right)^{*}}}}},} & (89) \\{{{C_{p^{\prime},\overset{\sim}{p},\hat{j},{\hat{j}}^{\prime}}^{{AC},{(4)}}\left( {{k_{\sigma};{\hat{}}_{D}},j,j^{\prime}} \right)} = {{Ae}^{i\; \kappa}W^{k_{d}}_{c}{\sum\limits_{ \in {\mathbb{Z}}}{_{_{D^{\prime}} - k_{d}}^{f,d}{\Psi^{AC}\left\lbrack {{;\left\{ ^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}},j,j^{\prime}} \right\rbrack} \times \left( {W^{k_{\sigma}_{c}}_{{\hat{}}_{D} - k_{n}}^{,d}W_{\mathcal{M}_{0}}^{- {\hat{j}}^{\prime}}_{_{D}^{(\hat{j})} - k_{d}^{(j)}}^{f,d}_{{{p\;}^{\prime}N^{\prime}M^{\prime}},{{- p}\; N\; M}}^{f,d}\frac{X_{\rho}^{(j)}}{\sqrt{N_{1}^{\prime}}}{\sum\limits_{n^{\prime} \in I_{P}^{(j)}}{_{{n^{\prime}M^{\prime}},{{- p}\; M}}^{f,d}X_{n^{\prime}}^{(j)}{G\lbrack \rbrack}}}} \right)^{*}}}}},} & (90)\end{matrix}$

each of which is called a correlator of type-3 with AC shifts and acorrelator of type-4 with AC shifts, respectively. In thetransmitter-signal-part (cf. (87)) of the two CCFs above, (i.e., theinput part of statistic g (13)) the TFSO of type 4 (or its frequencydual)

[MF716]

or

is inserted between the product of chip- and data-level shifting TFSO oftype 4 (or its frequency dual) at the transmitter (cf. υ[k;X^((i))], V[

;X^((i))] in (88))

[MF717]

_(qNM,q′N′M′) ^(d)·

_(m,M,m′M′) ^(d) or

_(q′N′M′,−qNM) ^(f,d)·

_(m′M′,−mM) ^(f,d)

and TFSO of type 3, representing the MC with shifts

[MF718]

(k _(d),

_(D)),

[MF719]

While, in the receiver-signal-part of the two CCFs (89), (90) (i.e., thesignal-to-be-detected part of statistic g(13)), for

[MF720]

-ary PSK modulation, there are

[MF721]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

different pairs of AC shifts

[MF722]

(k _(d) ^((j)),

_(D) ^((j)))

Then the associated estimated TFSOs of type 3 representing those ACs

[MF723]

(or their frequency dual ones

[MF724]

)

are defined. These TFSOs are inserted between the estimated TFSO of type1

[MF725]

(or the estimated TFSO of type 2

[MF726]

_({circumflex over (l)}) _(D) _(,−k) _(σ) ^(f,d))

for estimating the MC shifts

[MF727]

(k _(d),

_(D))

and the product of chip- and data-level shifting TFSOs of type 4 or itsfrequency dual at the receiver

[MF728]

_(pNM,p′N′M′) ^(d)·

_(n,M,ρ′M′) ^(d) or

_(p′N′M′,−pNM) ^(f,d)·

_(n′M′,−ρM) ^(f,d)

The transmitter/receiver system shown in FIGS. 12 and 13 that is capableof being used also as an efficient and high-resolution radar, equippedwith

[MF729]

-ary PSK communication is a typical example of multiplexingcommunication systems utilizing non-commutative delay and Doppler shiftsof the MC and AC parameter spaces of co-dimension 2, where thefundamental unit plane Θ′ of the delay and Dopler parameter space of theMC is exclusively divided, denoted as Θ^((i),′). So, it is obvious thatthe term “delay-Doppler space division multiplexing (dD-SDM)” [30],patent[6] is used to refer such a multiplexed system. In this system, inplace of transmitting

[MF730]

-ary data k, the transmitter sends a signal containing encoded integers

[MF731]

${\hat{j} = \left\lbrack \frac{\hat{k}}{\mathcal{M}_{0}} \right\rbrack},{\hat{j} = {\left\{ \frac{\hat{k}}{\mathcal{M}_{0}} \right\}.}}$

The integer j is recovered by using the PD due to the delay shift

[MF732]

k _(d) ^((ĵ))

and Doppler shift

[MF733]

_(D) ^((ĵ)),

located at around the center of Θ^((j∧),′) (cf. FIGS. 14,15). To embed(j, j′) into a signal, the transmitter firstly 2-D BPSK-modulates a chipwaveform by different 2-D PCs χ^((i)), secondly combines these modulatedsignals (signatures) in non-overlapped form by

[MF734]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right) - ^{(i)}$

-multiplexing, thirdly inputs it to the

[MF735]

ĵ

-th AC, and fourthly

[MF736]

₀

-PSK demodulates the output signal of the AC by the symbol

[MF737]

$\mathcal{M}_{0}\mspace{11mu} {ary}\mspace{14mu} {{\exp \left( \frac{{- {i2}}\; \pi {\hat{j}}^{\prime}}{\mathcal{M}_{0}} \right)}.}$

For [MF738]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

different values of

[MF739]

ĵ,

the transmitter requires

[MF740]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

parallel ACs and the set of those associated 2-D PCs

[MF741]

$\left\{ ^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}.$

While, at the receiver to detect the template from afourfold-multiplexed chip waveform, the receiver selects an estimated

[MF742]

ĵ

-th AC and constructs maximum-likelihood estimates of

[MF743]

(k _(d),

_(D)).

High [MF744]

-ary PSK communication at the transmitter is realized by

[MF745]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

-multiplexed low

[MF746]

₀

-ary PSK modulations based on using non-commutative AC shifts and 2-DPCs.

Such a multiplexed system, called “delay-Doppler space divisionmultiplexing (dD-SDM)” is a system by dividing the signal spaceS=[0,T_(s))×[0,F_(s)) into several sub-TFPs and assigning a 2-D PC toeach sub TFP for 2-D BPSK modulation. The use of 2-D PC results inadding a new axis perpendicular to the TFP as a third axis (see FIGS. 16and 17). Namely, this third axis represents a shift-axis, beinginherited the NCP from AC shifts of co-dimension 2. Therefore, thedD-SDM is radically different from the conventional division multipleaccess (DMA): TDMA, FDMA, CDMA, and Multi-carrier (MC)-CDMA. The spreadspectrum (SS) code used in the latter two that makes DMA easy by theorthogonality of the code-modulated rectangular chip pulse. The PULalgorithm based on the von Neumann's APT that enables us to jointlyestimate delay and Doppler with high precision, and is capable of beingused also for an efficient and high resolution radar. Such an M-ary PSKmodulation is a proposing and primitive technique in opticalcommunication because the notion of the AC with non-commutative shiftsis applicable to a “physical” optical fiber. Such a multiplexed system,called “delay-Doppler space division multiplexing (dD-SDM)” is a systemusing a non-commutative shift parameter space of co-dimension 2. Thishas a three-dimensionalized time-frequency space (TFS), layered by ACshifts (see FIGS. 16 and 17), in which the perpendicular co-ordinateaxis to the symbol space, time-frequency plane (TFP) of[0,T_(s))×[0,F_(s)), (which is identified with the unit plane of thetarget space), is a shift-axis with a scale of non-commutative ACshifts. Thus the dD-SDM is different from the conventional divisions ofsignal-TFP: TDMA and FDMA. The PUL algorithm based on APT that enablesus to estimate parameters with high-speed and high-precision is afundamental technique. Note that higher M-ary PSK modulation is apromising and primitive technique which is applicable to opticalcommunication because the AC with non-commutative shifts may be replacedby several kinds of physical transmission lines.

The references[30] and patent[6] regarded a multiple targets problemwith

[MF747]

{(k _(d,i),

_(D,i))}_(i=1) ^(N) ^(path)

as an application of Code-Division Multiple Target (CDMT)s and gave antarget-detection method using N_(path) 2-D PCs and TFSOs with shifts

[MF748]

{(k _(d,i),

_(D,i))}_(i=1) ^(N) ^(path) ,

together with the phase-cancellations by the product of two twiddlefactors

[MF749]

${W_{\mathcal{M}_{0}}^{\lambda \; \rho} \times W_{\mathcal{M}}^{j}},{\lambda_{\overset{\rightarrow}{p}} \in \left\{ {0,1,\ldots \;,{\mathcal{M}_{0} - 1}} \right\}},{j \in \left\{ {0,1,\ldots \;,{\left\lbrack \frac{\mathcal{M}}{\mathcal{M}_{0}} \right\rbrack - 1}} \right\}},{W_{\mathcal{M}} = {{\exp \left( \frac{{- i}\; 2\; \pi}{\mathcal{M}} \right)}.}}$

However, this method forces us to resolve small phase quantity

[MF750]

${\exp \left( \frac{{- i}\; 2\; \pi}{\mathcal{M}} \right)}.$

Thus when

[MF751]

≥16,

decoding errors arise due to phase noise and phase distortions.Furthermore, in the references above, without using both the TFSO oftype 3

[MF752]

representing MC shifts

[MF753]

(k _(d),

_(D))

and its estimated TFSO of type 4

[MF754]

,

[MF755]

-ary communication were done. Moreover, this approach was not suited tomultiple targets problems of a doubly delay-Doppler shifted signal andhas no estimation algorithm with guaranteed proof.To solve this, in accordance with an embodiment of the presentinvention, the inventor defines two new CCFs exploiting the NCP of ACshifts

[MF756]

c _(ρ′,{right arrow over (p)},ĵ,ĵ′) ^(AC,(3))(

_(μ) ;{circumflex over (k)} _(d) ,j,j′),C_(ρ,{right arrow over (p)},ĵ,ĵ′) ^(AC,(4))(k _(σ);

_(D) ,j,j′)

respectively defined by (89), (90), in which several PDs are estimatedand canceled out, precisely and rigorously. It is not easy to recover

[MF757]

d′ _({right arrow over (q)})·

embedded in TD-CE in (87)

[MF758]

$\psi^{AC}\left\lbrack {{k;\left\{ \chi^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}},j,j^{\prime}} \right\rbrack$

(or FD-CE [MF759]

$\left. {\Psi^{AC}\left\lbrack {{;\left\{ \chi^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}},j,j^{\prime}} \right\rbrack} \right)$

by

[MF760]

₀-ary PSK

The detail of cumbersome calculations of (89), (90) are omitted here.However, the two CCFs for high

[MF761]

-ary PSK demodulation are required by using

[MF762]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

2-D PCs [MF763]

$\left\{ \left( {X^{(j)},X^{{(j)},\prime}} \right\}_{j = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack} \right.$

and replacing

[MF764]

(k _(d),

_(D))⇒(k _(d) +k _(d) ^((j)),

_(D)+

_(D) ^((j))),d _({right arrow over (p)})

d′ _({right arrow over (p)})

at the transmitter and

[MF765]

({circumflex over (k)} _(d),

_(D))⇒({circumflex over (k)} _(d) +k _(d) ^((ĵ)),

_(D)+

_(D) ^((ĵ))),d _({right arrow over (q)}) ⇒d′ _({right arrow over (q)})

,(N,N′)⇒(N ₁ ,N′ ₁)

at the receiver. This replacement is accompanied by the occurrence ofseveral new PDs. Hence one should replace 2 variables for MLE operationof 2 variables

[MF766]

argmax_(ρ′{right arrow over (p)})

(or

[MF767]

argmax_(ρ,{right arrow over (p)}))

by the one of 4 variables

[MF768]

argmax_(ρ′{right arrow over (p)},ĵ,ĵ′)

(or

[MF769]

argmax_(ρ,{right arrow over (p)},ĵ,ĵ′)),

where

[MF770]

ρ′∈I _(F) ^((ĵ))

(or

[MF771]

ρ∈I _(T) ^((ĵ))).

One can obtain generalized versions of (39), (44):

[MF772]

$\begin{matrix}{{c_{\rho^{\prime},\overset{\rightarrow}{p},\hat{j},\hat{j^{\prime}}}^{{A\; C},{(3)}}\left( {{_{\mu};{\hat{k}}_{d}},j,j^{\prime}} \right)} = {\frac{\begin{matrix}{A\; e^{i\; \kappa}W^{{({k_{d} - {\hat{k}}_{d}})}_{c}}} \\W_{\mathcal{M}_{0}}^{- {({j^{\prime} - {\hat{j}}^{\prime}})}}\end{matrix}}{\sqrt{{PP}^{\prime}}}{\sum\limits_{\overset{\rightarrow}{q}}{\frac{d_{\overset{\rightarrow}{q}}}{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack}{\sum\limits_{i = 1}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}{\frac{X_{\rho^{\prime}}^{{(\hat{j})},\prime,*}}{N_{1}\sqrt{N_{1}^{\prime}}}{\sum\limits_{\substack{{{({m,m^{\prime}})} \in I^{(i)}}, \\ n \in I_{}^{G}}}{X_{m}^{(i)}X_{m^{\prime}}^{{(i)},\prime}X_{n}^{{(\hat{j})},*} \times {\quad{{\theta_{gg}\left\lbrack {{\left( {{\hat{k}}_{d} - k_{d}} \right) + \left( {{\hat{k}}_{d}^{(\hat{j})} - k_{d}^{(j)}} \right) + {\left( {p - q} \right){MN}_{1}} + {\left( {n - m} \right)M}},{\left( {_{\mu} - _{D}} \right) + \left( {{\hat{}}_{D}^{(\hat{j})} - _{D}^{(j)}} \right) + {\left( {p^{\prime} - q^{\prime}} \right)M^{\prime}N_{1}^{\prime}} + {\left( {\rho^{\prime} - m^{\prime}} \right)M^{\prime}}}} \right\rbrack} \times W^{\frac{1}{2}{({{k_{d}_{\mu}} - {{\hat{k}}_{d}_{D}} - {{\tau_{0}{\lbrack k_{\sigma}\rbrack}}{({_{D}^{(j)} + _{D}^{(\hat{j})}})}} + {{({k_{d}^{(j)} + k_{d}^{(\hat{j})}})}{v_{0}{\lbrack _{\mu}\rbrack}}} - {k_{d}^{(\hat{j})}_{d}^{(j)}} + {k_{d}^{(j)}_{d}^{(\hat{j})}}})}} \times W^{\frac{1}{2}{({{2{MN}_{1}q\; {{\overset{\sim}{v}}_{0}{\lbrack{_{\mu},\hat{j}}\rbrack}}} - {2M^{\prime}N^{\prime}q^{\prime}{\overset{\sim}{\tau_{0}}{\lbrack{k_{\sigma},\hat{j}}\rbrack}}}})}} \times {W^{\frac{1}{2}{({{{({{m\; \rho^{\prime}} - {m^{\prime}n}})}M\; M^{\prime}} - {{({m^{\prime} + \rho^{\prime}})}M^{\prime}{\overset{\sim}{\tau_{0}}{\lbrack{k_{\sigma},\hat{j}}\rbrack}}} + {{({m + n})}M\; {\overset{\sim}{v_{0}}{\lbrack{_{\mu},\hat{j}}\rbrack}}}})}}.}}}}}}}}}}} & (91) \\{{C_{\rho,\overset{\rightarrow}{p},\hat{j},\hat{j^{\prime}}}^{{A\; C},{(4)}}\left( {{k_{\sigma}\text{:}{\hat{}}_{D}},j,j^{~\prime}} \right)} = {\frac{\begin{matrix}{A\; e^{i\; \kappa}W_{\mathcal{M}_{0}}^{\hat{j}}W^{{- {\tau_{0}{\lbrack k_{\sigma}\rbrack}}}_{c}}} \\W_{\mathcal{M}_{0}}^{- {({j^{\prime} - \hat{j^{\prime}}})}}\end{matrix}}{\sqrt{{PP}^{\prime}}}{\sum\limits_{\overset{\rightarrow}{q}}{\frac{d_{\overset{\rightarrow}{q}}}{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack}{\sum\limits_{i = 1}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}{\frac{X_{\rho^{\prime}}^{{(\hat{j})},*}}{N_{1}^{\prime}\sqrt{N_{1}}}{\sum\limits_{\substack{{{({m,m^{\prime}})} \in I^{(\hat{i})}}, \\ n^{\prime} \in I_{F}^{G}}}{X_{m}^{(i)}X_{m^{\prime}}^{{(i)},\prime}X_{n}^{{(\hat{j})},\prime,*} \times {\Theta_{GG}\left\lbrack {{\left( {{\hat{}}_{D} - _{D}} \right) + \left( {{\hat{}}_{D}^{(\hat{j})} - _{D}^{(j)}} \right) + {\left( {p^{\prime} - q^{\prime}} \right)M^{\prime}N_{1}^{\prime}} + {\left( {n^{\prime} - m^{\prime}} \right)M^{\prime}}},{{- \left( {k_{\sigma} - k_{d}} \right)} - \left( {{\hat{k}}_{d}^{(\hat{j})} - k_{d}^{(j)}} \right) - {\left( {p - q} \right){MN}_{1}} - {\left( {\rho - m} \right)M}}} \right\rbrack} \times W^{\frac{1}{2}{({{k_{d}{\hat{}}_{D}} - {k_{\sigma}_{D}} - {{\tau_{0}{\lbrack k_{\sigma}\rbrack}}{({_{D}^{(j)} + _{D}^{(\hat{j})}})}} + {{v_{0}{\lbrack{\hat{}}_{D}\rbrack}}{({k_{d}^{(j)} + k_{d}^{(\hat{j})}})}} - {k_{d}^{(\hat{j})}_{D}^{(j)}} + {k_{d}^{(j)}_{D}^{(\hat{j})}}})}} \times W^{\frac{1}{2}{({{2{MN}_{1}q\; {{\overset{\sim}{v}}_{0}{\lbrack{_{\mu},\hat{j}}\rbrack}}} - {2M^{\prime}N_{1}^{\prime}q^{\prime}{{\overset{\sim}{\tau}}_{0}{\lbrack{k_{\sigma},\hat{j}}\rbrack}}}})}} \times {W^{\frac{1}{2}{({{{({{mn}^{\prime} - {m^{\prime}\rho}})}{MM}^{\prime}} - {{({m^{\prime} + n^{\prime}})}M^{\prime}{{\overset{\sim}{\tau}}_{0}{\lbrack{k_{\sigma},\hat{j}}\rbrack}}} + {{({m + \rho})}M\; {{\overset{\sim}{v}}_{0}{\lbrack{_{\mu},\hat{j}}\rbrack}}}})}}.}}}}}}}}} & (92)\end{matrix}$

where

[MF773]

{tilde over (τ)}₀[k _(σ) ,ĵ]=k _(σ) −k _(d) +k _(d) ^((ĵ)) −k _(d)^((j)),{tilde over (ν)}₀[

_(μ) ,ĵ]=

_(μ)

_(D)+

_(D) ^((j))−

_(D) ^((j)),

[MF774]

I _(T) ^((ĵ)) ,I _(F) ^((ĵ))

denote the sets of chip TD- and FD-addresses n,n′, associated with thej{circumflex over ( )}-th sub-TFP

[MF775]

S ^((ĵ)),

and the product of the address sets

[MF776]

I ^((ĵ)) =I _(T) ^((ĵ)) ×I _(F) ^((ĵ)).

The ambiguity function (AF)s: θ_(gg)[τ,υ] and Θ_(GG)[τ,υ] respectivelyin (91) and (92) with Gaussian functions g,G are separable and decayexponentially in terms of τ,υ. This well-known property and the disjointchip address sets

[MF777]

I ^((ĵ)) ,I _(T) ^((ĵ)) ,I _(F) ^((ĵ))

and the fact that

[MF778]

(k _(d) ^((j)),

_(D) ^((j)))

is at around the center of Θ^((j)) tell us that among the first andsecond arguments of the AF all of the terms relating to (p−q)MN₁;(p′−q′)M′N₁′ with p′≠q′, p≠q and

[MF779]

(m,m′)∈I ^((i)) ,n,ρ∈I _(T) ^((ĵ)) ,n′,ρ′∈I _(F) ^((ĵ))

with i≠j^({circumflex over ( )}) can be neglected. Thus, the terms withp′=q′, p=q and

[MF780]

i=ĵ

remain and thus only the term equal to

[MF781]

{right arrow over (p)}

is selected from the summation

[MF782]

$\sum\limits_{\overset{\rightarrow}{q}}$

and

[MF783]

ĵ=j

is identified via the functions τ^(˜)₀[k_(σ),j^({circumflex over ( )})],ν^(˜) ₀[

_(μ),j^({circumflex over ( )}′)] with k_(σ)k_(d),

_(μ)≃

_(D).This is the reason why the transmitter divides exclusively Θ, 2-D BPSKmodulates a chip waveform by 2-D PCs χ^((i)), and combines thesemodulated signatures in non-overlapped form to realize

[MF784]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

-multiplexing based on

[MF785]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

parallel ACs that are intended to introduce non-commutative shifts.While

[MF786]

ĵ′=j′

is identified by the PD cancellation term

[MF788]

.

The two CCFs above use independent 2-D PCs

[MF789]

$\left\{ \chi^{(i)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}$

and PDs due to non-commutative AC shifts

[MF790]

(k _(D) ^((ĵ)),

_(D) ^((ĵ)))

as functions of an estimated integer

[MF791]

$\hat{j} = \left\lbrack \frac{\hat{k}}{\mathcal{M}_{0}} \right\rbrack$

for an encoded integer

[MF792]

$j = \left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack$

of an integer k.The TD- and FD-CCFs provide us with a decoding method using PUL-basedMLEs. This method can guarantee the phase resolution at least

[MF793]

$W_{\mathcal{M}_{0}} = {\exp \left( {- \frac{i\; 2\; \pi}{\mathcal{M}_{0}}} \right)}$

and hence is resistant to phase noise.

It is natural to precisely estimate delay and Doppler for radar andestablish synchronization of communication systems before encoding anddecoding

[MF794]

-ary PSK data. Since one should confront a realistic situation that atransmitted signal through the MC with shifts

[MF795]

(k _(d),

_(D))

is processed at the receiver, we need two CCFs with AC shifts in (91)and in (92). Such receivers when the switches are on upward in FIGS.12,13 work as a synchronizer (i.e., acquisition and tracking) when theswitches are on the downward side, combine a synchronizer and a decoderfor

[MF796]

-ary PSK communication, and work as a radar equipped with datatransmission. This system is an amplitude-shift-keying(ASK)-free

[MF797]

-ary PSK communication and provides a secret wireless communicationsystem whose secret keys are 2-D PCs

[MF798]

$\left\{ {\chi^{(i)} = \left( {X^{(i)},X^{{(i)},\prime}} \right)} \right\}_{i = 0}^{\lbrack\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\rbrack}.$

The main reason that receivers employ ASK-free modulation is twofold:(1) adaptive and dynamic estimation of the attenuation factor Ae^(iκ);(2) a simplification of a decoder. Estimating and Canceling out PDsenable us to realize high

[MF799]

-ary PSK communication as an efficient use of radio resources. Note thatthe use of

[MF800]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

2-DPCs implies

[MF801]

$\left( {\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1} \right)$

-user multiplexed communication.

<8.3 Signal Processing Exploiting Multi-Dimensional Non-CommutativeSpatial-Spatial Frequency Shifts>

Daughman [33, 34] gave a 2-D version of Gabor elementary functions

[MF802]

g _(m,m′)(t)

g(t−mT _(c))e ^(i2πm′F′) ^(c) ^((t−mT) ^(c) ⁾ ,t∈

and pointed out that 2-D Gabor representation(to be defined below) isuseful for image analysis and segmentation. Before discussing Daughmanstudy, one should consult the Heisenberg group theory reviewed by Howe[5].

For [MF803]

t,τ,ν∈

^(n)

define two shift operators

[MF804]

T _(τ) f(t)

f(t−τ),M _(ν) f(t)

e ^(i2πν·t) f(t).  (93)

where ν·t denotes the inner product. For

[MF805]

z∈

introduce the scalar operator

[MF806]

S _(z) x(t)

zx(t).

Then the set [MF807]

{M _(ν) T _(τ) S _(z):ν,τ∈

^(n) ,z∈

}

is a group of unitary operators on

[MF808]

L ²(

).

Consider the set [MF809]

H=

^(n)×

^(n)×

,

called the (reduced) Heisenberg group of degree n and define on it a lawof combination

[MF810]

(ν₁,

₁ ,z ₁)(ν₂,

₂ ,z ₂)=(ν₁+ν₂,

₁+

₂ ,e ^(−2πiν) ² ^(·τ) ¹ z ₁ z ₂).  (94)

Define a map ρ, i.e., a faithful unitary representation of H and anautomorphism r of H by the rule

[MF811]

$\begin{matrix}\left. \begin{matrix}{\rho \text{:}\left( {v,\tau,z} \right)} & \rightarrow & {{{M_{v}T_{r}S_{z}},}\mspace{79mu}} \\{\mspace{11mu} {r\left( {v,\tau,z} \right)}} & \rightarrow & {\left( {{- \tau},v,{e^{2\pi \; i\; {\tau \cdot v}}z}} \right).}\end{matrix} \right\} & (95)\end{matrix}$

Then it is immediate to verify that the FT, denoted as {circumflex over( )} intertwines the two representations of H

[MF812]

ρ and ρor

with each other. That is, [5]

[MF813]

{circumflex over ( )}ρ(h)=ρ(r(h)){circumflex over ( )}.  (96)

Daughman[35] pointed out that the model of the 2-D receptive fieldprofiles encountered experimentally in cortical simple cells, whichcaptures their salient tuning properties of spatial localization,orientation selectivity, spatial frequency selectivity, and quadraturephase relationship, is the parameterized family of “2-D Gabor filters”.In terms of a space-domain (SD) impulse response function (or a complexGabor function) g(x) with SD variables

[MF814]

x=(x,y)

and its associated 2-D FT, i.e., spatial frequency-domain (SFD) function

[MF815]

ĝ(u)=

²[g(x)]

with SFD variables

[MF816]

u=(u,υ),

the general functional form of 2-D Gabor filter family is specified in[35, 36]

[MF817]

$\begin{matrix}\left. \begin{matrix}{{{{g(x)} = {{Ke}^{- {\pi {\lbrack{{a^{2}{({x - x_{0}})}}_{r}^{2} + {b^{2}{({y - y_{0}})}}_{t}^{2}}\rbrack}}} \cdot e^{i{({{2{\pi {({{u_{0}x} + {v_{0}y}})}}} + P})}}}},}\mspace{135mu}} \\{{{\hat{g}(u)} = {\frac{K}{ab}e^{- {\pi {\lbrack{{{({u - u_{0}})}_{r}^{2}a^{- 2}} + {{({v - v_{0}})}_{t}^{2}b^{- 2}}}\rbrack}}}}},e^{i{({{{- 2}{\pi {({{x_{0}{({u - u_{0}})}} + {y_{o}{({v - v_{0}})}}})}}} + P})}}}\end{matrix} \right\} & (97)\end{matrix}$

where P denotes the phase of the sinusoid such that the dc component iszero, and

[MF818]

(x−x ₀)_(r),(y−y ₀)_(r)

stand for a clockwise rotation of the operation by θ such that

[MF819]

$\begin{matrix}\left. \begin{matrix}{{{\left( {x - x_{0}} \right)_{r} = {{\left( {x - x_{0}} \right)\mspace{14mu} \cos \mspace{14mu} \theta} + {\left( {y - y_{0}} \right)\mspace{14mu} \sin \mspace{14mu} \theta}}},}\mspace{20mu}} \\{\left( {y - y_{0}} \right)_{r} = {{{- \left( {x - x_{0}} \right)}\mspace{14mu} \sin \mspace{14mu} \theta} + {\left( {y - y_{0}} \right)\mspace{14mu} \cos \mspace{14mu} {\theta.}}}}\end{matrix} \right\} & (98)\end{matrix}$

The SD function g(x), SFD function

[MF820]

ĝ(u)

are perfectly symmetrical and have their shifts

[MF821]

x ₀=(x ₀ ,y ₀), u ₀=(u ₀ ,v ₀)

at which 2-D Gaussian envelope has the peak value. An important propertyof the family of 2-D Gabor filters is their achievement of thetheoretical lower bound of joint uncertainty in the two conjoint domainsof x and u. Defining uncertainty in each of the four variables by thenormalized second moments

[MF822]

√{square root over (σ_(x) ²)},√{square root over (σ_(y) ²)},√{squareroot over (σ_(u) ²)},√{square root over (σ_(v) ²)}

about the principle axes, a fundamental uncertainty principle exists[35,36]:

[MF823]

√{square root over (σ_(x) ²)}√{square root over (σ_(u) ²)}≥¼π, √{squareroot over (σ_(y) ²)}√{square root over (σ_(v) ²)}≥¼π  (99)

and the lower bound of the inequality is achieved by the family of 2-DGabor filters (97). where

[MF824]

u ₀=(u ₀ ,v ₀),{right arrow over (α)}=(α,β)

are modulation, scaling parameters of Gaussian.

He proposed a simple neural network (NN) architecture for findingoptimal coefficient values in arbitrary 2-D signal transforms which ingeneral might be neither complete nor orthogonal as follows.

Consider some discrete 2-D signal I[x], say, an image supported on[256×256] pixels in x, which one wants to analyze or compress byrepresenting it as a set of expansion coefficients {a_(i)} on some setof 2-D elementary functions

[MF825]

{g[x;{right arrow over (θ)} _(i)]}, {right arrow over (θ)}_(i)=(x _(0i)u _(0i),{right arrow over (α)}_(i)),x _(0i)=(x _(0i) ,y _(0i)), u_(0i)=(u _(0i) ,v _(0i)),α_(i)=({right arrow over (α)}_(i),β_(i)).

The attempt to represent I[x] either exactly or in some optimal sense byprojecting it onto a chosen set of vectors {g[x;θ^(→) _(i)]} is reducedto finding projection coefficients {a_(i)} such that the resultantvector H[x]

[MF826]

$\begin{matrix}{{H\lbrack x\rbrack} = {\sum\limits_{i = 1}^{n}\; {a_{i}{g\left\lbrack {x;{\overset{\rightarrow}{\theta}}_{i}} \right\rbrack}}}} & (100)\end{matrix}$

minimizes the squared norm of the difference-vector E=∥I[x]−H[x]∥². If agiven image I[x] is regarded as a vector in an n-D vector space (e.g.n=65 536), then the optimal coefficients

[MF827]

{a _(i)}_(i=1) ^(n)

is determined by a system of n simultaneous equations in n unknowns; butit is impractical to solve this huge system. So he proposed NNarchitectures with connection strength

[MF828]

{g[x;{right arrow over (θ)} _(i)]}

and input image I[x]. 2-D Gabor representations for image analysis andcompression, edge detections, feature detectors etc. have been proposed.[37, 38, 39]In this disclosure, the inventor points out that the 2-D Gabor expansion(100) can be regarded as a conjoint 2-D spatial/spatial frequency (S/SF)representation using non-commutative shifts (x_(0i),u_(0i)) in terms of2-D Gabor functions

[MF829]

{g[x;{right arrow over (θ)} _(i)]}

with shifts (x_(0i),u_(0i)): According to an important precedent for 1-Dsignal processings exploiting the non-commutativity of time-frequencyshifts, presented in the previous sections in this disclosure, it isnatural to consider SD representation (100) and its associated SFDrepresentation, symmetrically. To do it, let 2-D FT of I[x] and H[x]

[MF830]

Î[u]=

⁽²⁾[I[x]], Ĥ[u]=

⁽²⁾[H[x]].

Then the conjoint spatial/spatial frequency representation, in place of(100), is defined as

[MF831]

$\begin{matrix}{{{H\lbrack x\rbrack} = {\sum\limits_{i = 1}^{n}\; {a_{i}{g\left\lbrack {x;{\overset{\rightarrow}{\theta}}_{i}} \right\rbrack}}}},{{\hat{H}\lbrack u\rbrack} = {\sum\limits_{i = 1}^{n}\; {a_{i}^{\prime}{{\hat{y}\left\lbrack {u;{\overset{\rightarrow}{\theta}}_{i}} \right\rbrack}.}}}}} & (101)\end{matrix}$

and let ε,ε′ be two subspaces of the Hilbert space. Suppose that

[MF832]

I[x]∈ε,Î[u]∈ε′.

The squared norms of errors

[MF833]

∥I[x]−H[x]∥² , ∥I[u]−H[u]∥²

are minimized by the use of two orthogonal projection operator (PO)sprojecting onto ε_(i), ε′_(i) (defined below), respectively

[MF834]

{P _(g(x;{right arrow over (θ)}) _(i) ₎ ,P _(ĝ(u,{right arrow over (θ)})_(i) ₎}_(i=1) ^(n).

The projection theorem [27] tells us that

[MF835]

I[x],Î[u]

possess their unique decomposition

[MF836]

$\begin{matrix}\left. \begin{matrix}{{{I(x)} = {{\sum\limits_{i = 1}^{n}\; {P_{g{({x;{\overset{\rightarrow}{\theta}}_{i}})}}{I(x)}{g\left( {x;{\overset{\rightarrow}{\theta}}_{i}} \right)}}} + {g^{\bot}\left( {x;{\overset{\rightarrow}{\theta}}_{i}} \right)}}},{{P_{g{({x;{\overset{\rightarrow}{\theta}}_{i}})}}{I(x)}} = \frac{\Sigma_{x}{{g^{*}\left( {x;{\overset{\rightarrow}{\theta}}_{i}} \right)} \cdot {I(x)}}}{\Sigma_{x}{{g\left( {x;{\overset{\rightarrow}{\theta}}_{i}} \right)}}^{2}}},} \\{{{\hat{I}(u)} = {{\sum\limits_{i = 1}^{n}\; {P_{\hat{g}{({u;{\overset{\rightarrow}{\theta}}_{i}})}}{\hat{I}(u)}{\hat{g}\left( {u;{\overset{\rightarrow}{\theta}}_{i}} \right)}}} + {{\hat{g}}^{\bot}\left( {u;{\overset{\rightarrow}{\theta}}_{i}} \right)}}},{{P_{\hat{g}{({u;{\overset{\rightarrow}{\theta}}_{i}})}}{\hat{I}(u)}} = \frac{\Sigma_{u}{{{\hat{g}}^{*}\left( {u;{\overset{\rightarrow}{\theta}}_{i}} \right)} \cdot {\hat{I}(u)}}}{\Sigma_{u}{{\hat{g}\left( {u;{\overset{\rightarrow}{\theta}}_{i}} \right)}}^{2}}},}\end{matrix} \right\} & (102)\end{matrix}$

where g^(⊥)(x;θ^(→) _(i)), g^({circumflex over ( )}⊥) (u; θ^(→) _(i))are orthogonal complements of g(x; θ^(→) _(i)),g^({circumflex over ( )})(u; θ^(→) _(i)), respectively. Therefore it isimportant to choose a set of templates

[MF837]

{g(x;{right arrow over (θ)} ₂),ĝ(u;{right arrow over (θ)} _(i))}_(i=1)^(n),

For example, let ε_(i) be the set of all f∈

₂(Z) that are (L₁Δx×L₂Δy) space-limited (SL) signals and let ε′_(i) bethat of (L₁Δu×L₂Δv) spatial frequency-limited (SFL) signals. Supposethat

[MF838]

g(x;{right arrow over (θ)} _(i))∈ε_(i) and ĝ(u,{right arrow over (θ)}_(i))∈ε′_(i).

Then, [MF839]

P _(g(x;{right arrow over (θ)}) _(i) ₎ ,P _(ĝ(u;{right arrow over (θ)})_(i) ₎

are POs projecting onto ε_(i),ε′_(i); respectively. Apply the vonNeumann's APT to the two POs and we use 2 2×2-D localization operators(LOs) defined by

[MF840]

P _(g(x;{right arrow over (θ)}) _(i) ₎

^(−1,(2),d) P _(ĝ(u;{right arrow over (θ)}) _(i) ₎

^((2),d) ,P _(ĝ(u,{right arrow over (θ)}) _(i) ₎

^((2),d) P _(g(x;{right arrow over (θ)}) _(i) ₎

^(−1,(2),d)  (103)

then one can get optimized expansion coefficients given as

[MF841]

a _(i) ˜P _(g(x;{right arrow over (θ)}) _(i) ₎ I(x),a′ _(i) ˜P_(ĝ(u;{right arrow over (θ)}) _(i) ₎ I(u),a _(i) ˜a′ _(i), 1≤i≤n.  (104)

Such 2-D LOs select a limited 4-D cube

[MF842]

[L ₁ Δx×L ₂ Δy]×[L′ ₁ Δu×L′ ₂ Δv]⊂

⁴

around the peak address (x_(0i),u_(0i)) in the 4-D spatial/spatialfrequency plane and filter out the rest. where

[MF843]

g ^(⊥)(x;{right arrow over (θ)} _(i))∈ε_(i) ^(⊥) and ĝ ^(⊥)(u;{rightarrow over (θ)} _(i))∈ε_(i) ^(⊥,t).

[MF844]

ε_(i) ^(⊥),ε_(i) ^(⊥,)′

are orthogonal complements of

[MF845]

ε_(i),ε′_(i).

Contrary to Howe's shift operators (93), for the SD- and SFD-signals

[MF846]

z(x),{circumflex over (z)}(u)

the inventor introduces “half-shifts” for the symmetrical property

[MF847]

e ^(−πu) ⁰ ^(·x) ⁰ ,e ^(iπx) ⁰ ^(·u) ⁰ ,x ₀=(x ₀ ,y ₀), u ₀=(u ₀ ,v ₀)∈

²

and uses 2×2-D von Neumann's symmetrical spatial-spatial frequency shiftoperator (SSFSO)s with those half-shifts, defined by

[MF848]

$\begin{matrix}{{{_{x_{0},u_{0}}^{(2)}{z(x)}} = {{z\left( {x - x_{0}} \right)} \cdot e^{i\; 2\pi \; {u_{0} \cdot {({x - \frac{x_{0}}{2}})}}}}},{{_{u_{0},{- x_{0}}}^{{(2)},f}{\hat{z}(u)}} = {{\hat{z}\left( {u - u_{0}} \right)} \cdot e^{{- i}\; 2\pi \; {x_{0} \cdot {({u - \frac{u_{0}}{2}})}}}}}} & (105)\end{matrix}$

for the 2-D Gabor function with peak-address x₀=(x₀, y₀), u₀=(u₀, v₀)∈R²in 2×2-D SD-SFD space

[MF849]

g(x;{right arrow over (θ)} _(i))

and its FT [MF850]

ĝ(u;{right arrow over (θ)} _(i))=

⁽²⁾[g(x;{right arrow over (θ)} _(i))].

Thus, the inventor can rewrite the Gabor functions as

[MF851]

$\begin{matrix}\left. \begin{matrix}{{\left. {{g\left( {x;{\overset{\rightarrow}{\theta}}_{i}} \right)} = {_{x_{0i},u_{0i}}^{(2)}\left\lbrack {e^{- {\pi {({{x^{2}\alpha_{i}^{2}} + {y^{2}\beta_{i}^{2}}})}}} \cdot e^{2i\; \pi \; {x \cdot u}}} \right)}} \right\rbrack \cdot e^{{- \pi}\; {{iu}_{0i} \cdot x_{0i}}}},} \\{{\hat{g}\left( {u;{\overset{\rightarrow}{\theta}}_{i}} \right)} = {{_{u_{0i},{- x_{0i}}}^{{(2)},f}\left\lbrack {e^{- {\pi {({{v^{2}\alpha_{i}^{- 2}} + {v^{2}\beta_{i}^{- 2}}})}}} \cdot e^{{- 2}i\; \pi \; {x \cdot u}}} \right\rbrack} \cdot {e^{\pi \; {{ix}_{0i} \cdot u_{0i}}}.}}}\end{matrix} \right\} & (106)\end{matrix}$

Consider the DFT of the above signals. Let (Δx,Δy) be the samplingintervals of 2-D space variables

[MF852]

x=(x,y)

with Cartesian lattice dimensions (L₁,L₂). Let

[MF853]

z[{right arrow over (k)}]

be a discrete SD function whose discrete space variable is

[MF854]

${\overset{\rightarrow}{k} = {\left( {k_{1},k_{2}} \right) \in {\mathbb{Z}}^{2}}},{k_{1} = \left\lfloor \frac{x}{\Delta \; x} \right\rfloor},{k_{2} = {\left\lfloor \frac{y}{\Delta \; y} \right\rfloor.}}$

Then the sampling interval of 2-D spatial frequencies

[MF855]

u=(u,v)

appropriate to the L₁, L₂ spatial lattice are given by

[MF856]

Δu=1/(L ₁ Δx),Δv=1/(L ₂ Δy)

and its discrete spatial frequency variable is given by

[MF857]

${\overset{\rightarrow}{} = {\left( {_{1},_{2}} \right) \in {\mathbb{Z}}^{2}}},{_{1} = \left\lfloor \frac{u}{\Delta \; u} \right\rfloor},{_{2} = {\left\lfloor \frac{v}{\Delta \; v} \right\rfloor.}}$

Using 2 twiddle factors

[MF858]

${W_{1} = e^{{- i}\frac{2\pi}{M}}},{W_{2} = e^{{- i}\sqrt[N]{2\pi}}}$

and 2-D DFT [MF859]

^((2),d)[⋅], and 2-D IDFT

^(−1,(2),d)[⋅],

one can obtain the relationships between an SFD function

[MF860]

Z[

]

and an SD function

[MF861]

z[{right arrow over (k)}]

[MF862]

$\begin{matrix}\left. \begin{matrix}{{{Z\left\lbrack \overset{\rightarrow}{} \right\rbrack} = {{\mathcal{F}^{{- 1},{(2)},d}\left\lbrack {z\left\lbrack \overset{\rightarrow}{k} \right\rbrack} \right\rbrack}\overset{def}{=}{\frac{1}{\sqrt{L_{1}L_{2}}}{\sum\limits_{k_{1},{k_{2} = 0}}^{{L_{1} - 1},{L_{2} - 1}}\; {{z\left\lbrack \overset{\rightarrow}{k} \right\rbrack}W_{1}^{k_{1}_{1}}W_{2}^{k_{2}_{2}}}}}}},{0 \leq _{i} \leq L_{i}},{i = 1},2,} \\{{{z\left\lbrack \overset{\rightarrow}{k} \right\rbrack} = {{\mathcal{F}^{{(2)},d}\left\lbrack {Z\left\lbrack \overset{\rightarrow}{} \right\rbrack} \right\rbrack}\overset{def}{=}{\frac{1}{\sqrt{L_{1}L_{2}}}{\sum\limits_{_{1},{_{2} = 0}}^{{L_{1} - 1},{L_{2} - 1}}\; {{Z\left\lbrack \overset{\rightarrow}{} \right\rbrack}W_{1}^{{- k_{1}}_{1}}W_{2}^{{- k_{2}}_{2}}}}}}},{0 \leq k_{i} \leq L_{i}},{i = 1},2.}\end{matrix} \right\} & (107)\end{matrix}$

Thus the SD function

[MF863]

z[{right arrow over (k)}]

(or the SFD function

[MF864]

Z[

])

has support

[MF865]

L ₁ Δx×L ₂ Δy

(or

[MF866]

L ₁ Δu×L ₂ Δv).

If the spacing of the peak-address x_(0i) (or u_(0i)) on the SD space(or SFD space) is

[MF867]

(M _(x) Δx,M _(y) Δy) or (M _(u) Δu,M _(v) Δv)

and the normalization condition

[MF868]

M _(x) Δx×M _(u) Δu=M _(y) Δy×M _(v) Δv=1

is imposed, then

[MF869]

L ₁ =M _(x) M _(u) , L ₂ =M _(y) M _(v).

If discrete space/spatial frequency shifts of

[MF870]

g(x;{right arrow over (θ)} _(i)),ĝ(u;{right arrow over (θ)} _(i))

are given as

[MF871]

${a_{i} = \left( {a_{i\; 1},a_{i\; 2}} \right)},{b_{i} = {\left( {b_{i\; 1},b_{i\; 2}} \right) \in {\mathbb{Z}}^{2}}},{a_{i\; 1} = \left\lfloor \frac{x_{0i}}{\Delta \; x} \right\rfloor},{a_{i\; 2} = \left\lfloor \frac{y_{0i}}{\Delta \; y} \right\rfloor},{b_{i\; 1} = \left\lfloor \frac{u_{0i}}{\Delta \; u} \right\rfloor},{b_{i\; 2} = {\left\lfloor \frac{v_{0i}}{\Delta \; v} \right\rfloor.}}$

then 2×2-D von Neumann's discrete SSFSOs become

[MF872]

$\begin{matrix}\left. \begin{matrix}{{{_{a_{i},b_{i}}^{{(2)},d}{z\left\lbrack \overset{\rightarrow}{k} \right\rbrack}} = {{z\left\lbrack {\overset{\rightarrow}{k} - \overset{\rightarrow}{a}} \right\rbrack}W_{1}^{- {b_{1}{({k_{1} - \frac{a_{1}}{2}})}}}W_{2}^{- {b_{2}{({k_{2} - \frac{a_{2}}{2}})}}}}},} \\{{_{b_{i},{- a_{i}}}^{{(2)},,d}{Z\left\lbrack \overset{\rightarrow}{} \right\rbrack}} = {{Z\left\lbrack {\overset{\rightarrow}{} - \overset{\rightarrow}{b}} \right\rbrack}W_{1}^{a_{1}{({_{1} - \frac{b_{1}}{2}})}}{W_{2}^{a_{2}{({_{2} - \frac{b_{2}}{2}})}}.}}}\end{matrix} \right\} & (108)\end{matrix}$

These 2×2-D symmetrical SSFSOs contain 2-D half shifts

[MF873]

${W_{1}^{\frac{b_{1}a_{1}}{2}} \cdot W_{2}^{\frac{b_{2}a_{2}}{2}}},{W_{1}^{- \frac{b_{1}a_{1}}{2}} \cdot W_{2}^{- \frac{b_{2}a_{2}}{2}}}$

respectively, as phase terms of SD and SFD signals

[MF874]

z[{right arrow over (k)}],Z[

].

Thus the SFSOs (108) may be useful for image analysis, featureextraction, and data compression because

[MF875]

(a _(i) ,b _(i)) i.e., (x _(0i) ,u _(0i))

are efficiently extracted through the inner product of the projectioncoefficients in (102). Note that image analysis which involves atemporal change is considered to be a subject of a signal processingexploiting three-dimensional NCP.

<9 Distinguishing Characteristics of the Invention, and of the Mannerand Process of Making and Using It>

Embodiments, mathematical formulae and figures, described in a writtendescription of the invention, and of the manner and process of makingand using it are merely just ones of examples of the process and/ormethod of the invention. The inventor can honestly say that some ofembodiments are realized examples of communication systems exploitingthe NCP of TFSs, that Gabor prospected deliberately in the 1952paper.[2]

The non-overlapping superposition of signals on the TFP necessarilyentails PDs due to the NCP of TFSs. Most of researchers in the field ofcommunication, however, has little or no understanding of TFSs with theNCP. Under such a situation, communication systems exploiting the NCP ofTFSs are expected to provide a new class of digital multiplexedcommunication systems.

Communication systems exploiting the NCP of TFSs, given in thisdisclosure, can be restated from the following five different points ofview.

(Viewpoint 1)

The symmetrical time and frequency shift operator (TFSO), defined andintroduced in the references (patent[1] and [26, 27, 30]) has theproperties: (i) its “half shifts”, i.e., TD- and FD-PDs that areembedded into TD- and FD-signatures makes clear shift's important rolein parameter estimation; (ii) the PD due to modulation and demodulationby the carrier l_(c), accompanied by the delay k_(d)

[MF876]

always arises; (iii) the fact that any PDs being embedded into a signalcan be compactly expressed in terms of powers of the twiddle factor

[MF877]

${W = e^{{- i}\frac{2\pi}{L}}},$

makes estimations of several PDs and canceling out the PDs easy.

(Viewpoint 2)

The TD- and FD-PC modulation, i.e., 2-D binary phase-shift-keying(BPSK)) is a well-known modulation technique. However if one can regardit as an example that TFSOs are available, then one can see that TD- andFD-templates are embedded in a wide-band signal modulated by 2D-PCs.

(Viewpoint 3)

The non-overlapped superposition of signals on the TFP is a conventionalmethod as an efficient use of radio resources. However, thissuperposition needs the use of TFSO that causes a data-level PD.Fortunately, such a PD plays an important role in parameter estimation.

(Viewpoint 4)

Contrary to conventional use of maximum likelihood (ML) TD-functionalsonly, the ML TD- and FD-functionals define arrays of TD- and FD-CCFs,respectively for detecting TD- and FD-templates. TD- and FD-CCFs defineorthogonal projection operator (PO)s P₃, P₄ (or P₁, P₂), respectivelyprojecting onto time-limited (TL) and band-limited (BL) spaces assubspaces of the Hilbert space. These POs provide a frame on which adeeper understanding of the APT can be built.

(Viewpoint 5)

Contrary to popular belief that a non-Nyquist Gaussian is useless,

The Alternative POs Representing a Conjunction of the Two POs

[MF878]

P ₈

^(−1,d) P ₄

^(d)

(or

[MF879]

P ₄

^(d) P ₃

^(−1,d))

becomes a localization operator in cooperation with the use of Gaussianhaving properties: self-dual of the FT, separable and exponentiallydecaying Gaussian's AF in terms of delay and Doppler. This leads thefact that arrays of TD- and FD-CCFs yield excellent receivers.The inventor makes all of the above viewpoints clear after publicationsin the patents and non-patent references. This disclosure features itsown novelty. All features of novelty are prescribed as follows

(Point 1 of Novelty)

Multiplexed communication systems exploiting the NCP of the TFSs on theTFP are provided. Also algorithms for evaluating PDs due to the TFSs andfor canceling out their PDs are given based on symmetrical TFSOs in (4),(24). Furthermore, such an operation pair of embedding- andcanceling-out-PDs is shown to be of crucial importance.

(Point 2 of Novelty)

Doppler- and delay-estimate likelihood TD- and FD-functionals,respectively enable us to get ML estimators of parameters and realizecanceling-out-PDs by their associated MLE correlators.

(Point 3 of Novelty)

It is shown that the classical BPSK modulation is causing PDs due to theNCP of its TFSs, but fortunately these PDs provide a clue aboutparameter estimation and signal reconstruction.

(Point 4 of Novelty)

Both the non-overlapped superposition of signals on the TFP andmodulation (or demodulation) for a baseband (or passband) signal,accompanied by a delay, as a conventional technique in communication,induce PDs due to the NCP of their TFSs. Fortunately,embedding-and-canceling-out-PD techniques play an important role indetecting a data-level signal.

(Point 5 of Novelty)

All of the above PDs are canceled out by TD- and FD-CCFs. These CCFsinduce TD- and FD-POs projecting onto the TL- and BL-signal spaces,respectively as subspaces of the Hilbert space. Applying the vonNeumann's alternative projection theorem (APT) to the two subspacesdefines the combined operator of the two POs, called a localizationoperator (LO) on the TFP.

(Point 6 of Novelty)

Youla's signal restoration method using the APT enables us to prove thatthe LO guarantees the convergence of the alternative parameter-updatingalgorithm, called PUL. This implies that the LO, as a function of data-and chip-level addresses, becomes a new kind of filters in place of aconventional sharp filter in the DSP and works well for a radar with nolimitation on the maxima of delay and Doppler.

(Point 7 of Novelty)

A 2-D signal, i.e., image is usually described by conjoint spatial(S)/spectral frequency (SF) representations, 2-D Gabor representationsin particular. However, many researchers using conventional techniquesare unaware that 2-D S and SF shift (SSFS)s, implicitly involved in 2-DGabor functions induce unexpectedly PDs due to their NCP like 1-D TFSs.The inventor treats an S domain (SD) signal and an SF domain (SFD)signals symmetrically and introduces conjoint S/SF representations ofdimension 2×2 in terms of 2-D Gabor functions and their 2-D FTs, definedas (101), in place of the problem of representing image in terms of 2-DGabor functions (100).Such a new symmetrical conjoint problem of representing image (101) canbe solved by the SD and SFD decomposition (102) using orthogonalprojection operator (PO)s projecting onto SD- and SFD-limited spaces assubspaces of the Hilbert space, respectively defined as

[MF880]

P _(g(x;{right arrow over (θ)}) _(i) ₎ ,P _(ĝ(u;{right arrow over (θ)})_(i) _().)

2-D Gabor functions are defined using symmetrical space and spatialfrequency shift operator (SSFSO)s (105) (or (108)). Furthermore, thecombined 2×2-D localization operator (LO)s (103) of the two POs that thevon Neumann's APT can be applied provide the basics of 2-D signalprocessing theory exploiting the NCP of SSFSs for purposes such as imageanalysis, feature extraction, and data compression.

<<Example 1 of Embodiments of Transmitter-Receiver System>>

An example as an embodiment of transmitter-receiver systems based on theabove-described theoretical aspects is given together with citedfigures.

According to an embodiment of the present invention, FIG. 18 shows ablock-diagram representing the transmitter-receiver system with thenumber 1 attached to its surrounding dotted line. The communicationsystem with the number 1 comprises the transmitter apparatus 100 and thereceiver apparatus 200. This transmitter-receiver system with the number1 is used for both a radar and a communication system (e.g.,transmitter-receiver system to convey voice data and or image data).

(The Transmitter Apparatus 100)

The transmitter apparatus 100 is an apparatus that executes a programimposed on a transmitter as described in the above description.

As shown in FIG. 18, the transmitter apparatus 100 comprises anacquisition part of transmitting data 101, a generator part of atransmitting signal 102, and a transmitter part 103 as an example.

(The Acquisition Part of Transmitting Data 101)

The acquisition part of transmitting data 101 makes acquisition of datato be transmitted; The transmitting data is, e.g., acoustic data, imagedata, and may be text data such as any digitized data.

If the transmitter-receiver system with the number 1 is used for radar,then the transmitting data may be a radar pulse wave.

(The Generator Part of a Transmitting Signal 102)

The generator part of a transmitting signal 102 performs signalprocessing transmitting data obtained by the acquisition part oftransmitting data 101 and generates a transmitting signal.

The generator part of a transmitting signal 102 comprises SFBs as shownin FIGS. 4-7 or at least one of them as an example.

As above mentioned, an example of processing for generators of a signalto be transmitted is the modulation by TD- and FD-phase code (PC)s ofperiods N,N′, i.e., 2-D BPSK modulation.

Examples of a transmitting signal generated by the generator part of asignal to be transmitted 102 are e.g., as mentioned above

[MF881]

-   -   TD, FD pulse waveforms: g[k], G[        ],    -   TD, FD templates: u_(m′) ⁽³⁾[k; X], U_(m) ⁽⁴⁾[        ; X′],    -   TD, FD signatures: v[k; χ], V[        ; χ],    -   TD, FD signals to be transmitted: s[k; χ], S[        ; χ],    -   CE of s[k; χ], FT of ψ[k; χ]:ψ[k; χ], Ψ[        ; χ],        The inventor can enumerate processing, described in the        subsection <4. signature waveforms and templates in TD and FD>        as concrete examples of processing in the generator part of a        transmitting signal.

The generator part of a signal to be transmitted with the number 102 maybe an apparatus that executes a program imposed on a transmitter asdescribed in subsection <7.1 transmitters for generating signature andradar signal>. Or the generator part of a signal to be transmitted withthe number 102 may be an apparatus that executes a program imposed on atransmitter apparatus as described in subsections <8.1 multiple targetdetection using CDMT> and <8.2 multiple target detection usingartificial delay-Doppler>.

(Transmitter Part 103)

The transmitter part 103 transmits a transmitter signal generated by thegenerator part of a signal to be transmitted with the number 102.

(Receiver Apparatus 200)

The receiver apparatus 200 is an apparatus that executes a programimposed on a receiver as described in the above description.

As shown in FIG. 18, the receiver apparatus 200 comprises the receiverpart 201, the shift estimation and received-data extraction parts 202,and the received-data decision part 203 as an example.

(Receiver Part 201)

The receiver part 201 receives a transmitter signal, transmitted by thetransmitter apparatus 100. Examples of a received signal that thereceiver part 201 receives are e.g., as mentioned above

[MF882]

received TD-, FD-signals:r[k;χ],R[

;χ],

(Shift Estimation and Received-Data Extraction Parts 202)

The shift estimation and received-data extraction parts 202, (simplycalled the estimation part) shifts a received signal by the receiverpart 201 by estimated shifts and extract the received data.

The shift estimation and received-data extraction parts 202 comprises atleast one of AFBs shown in FIGS. 8 to 10, as an example. The shiftestimation and received-data extraction parts 202 executes severalprograms e.g., as described in subsections <5. M-ary detection andestimation of TD and FD signals> and <6. TD and FD cross-correlationsfor parameter estimation>.

The shift estimation and received-data extraction parts 202 is anapparatus that executes a program as described in subsection <7.2 AFB;receivers and encoder design>. Moreover, the generator part of a signalto be transmitted 102 is an apparatus that executes a program asdescribed in subsection <8.2 High M-PSK communication based onexploiting non-commutative time-frequency shifts>.

Thus, the shift estimation and received-data extraction parts 202 is amethod for receiving a signal and executes a program according to theestimation step for estimating a time shift and a frequency shift from areceived signal (cf. FIG. 13 and (38), (43) (or (89), (90)),corresponding to switches 2-1, 2-2 on the upward (or downward) side byreferring to a non-commutative shift parameter space of co-dimension 2(cf. FIG. 17).

The generator part of a signal to be transmitted 102 executes a programaccording to the shift steps for time-frequency shifting the signal tobe transmitted (cf. FIG. 12 and (25), (27) (or (87), (88))),corresponding to switches 1-1,1-2 are on the upward (or downward) sideby referring the non-commutative shift parameter space of co-dimension 2(cf. FIG. 17).

The non-commutative shift parameter space of co-dimension 2 is a3-dimensionalized space version of the 2-D time-frequency plane with thetime- and frequency-coordinates by augmenting non-commutativetime-frequency shifts as a third coordinate (cf. FIG. 17).

As above mentioned, a pair of unknown delay and Doppler

[MF883]

(k _(d),

_(D))

is itself the pair of non-commutative shifts on the parameter space ofco-dimension 2 like the pair of AC shifts. The parameter space isrelated to the shift operations acting on a signal on the time-frequencyplane (TFP); Such a “shift plane” should be discriminated against theTFP (FIG. 16). Namely, to draw a distinction between four fundamentalarithmetic operations as well as differential and integral calculus interms of variables time t, frequency f and non-commutative shiftoperations for calculating phase terms, one should notice that the NCPof the shift operation is peculiarity of the proposed time-frequencyanalysis (e.g., the third coordinate of FIG. 17).

Several comments to the patent[6]-patent[8] are listed as follows.

a) The phase distortion (PD)

[MF884]

(or its discretized version

[MF885]

that the modulation and demodulation by the carrier f_(c) accompanied bythe delay t_(d) (or k_(d)) are causing was neglected in the abovepatents. While in this disclosure, this PD and the PD e^(iκ) of theattenuation factor Ae^(iκ) are jointly and co-operatively canceled out.The reason that the co-operative treatment of PDs is needed is asfollows; Firstly, each PD is not independent of each other; Secondly,the PD is propagated due to its group-theoretic property; Thus it is notan easy task to cancel out these PDs precisely. Consequently, theprocess in updating estimates of time and frequency shifts should beundertaken in co-operation with the phase cancellations (i.e.,simultaneous and parallel processing is necessary: e.g., (59)).

b) Time and frequency shifts in this disclosure are referred to asvariables of the parameter space of co-dimension 2, associated with twovariables: time and frequency on the signal's TFP, respectively. Thenon-commutative shift operations induce the PDs. While, time andfrequency offsets in communication are deviations from the pulse'speriod and the carrier, respectively.

(1) the estimated time and frequency offsets in patents[7, 8] and timeand frequency offsets in OFDM-based communication system in patents[7]are deviations from the period of a chip pulse and the carrier,respectively and correspond to the time shift t_(d) and the frequencyshift f_(D) in this disclosure, respectively. Namely, the above both oftwo patents can be regarded as conventional acquisition and trackingmethods for synchronisation because these are unaware of thenon-commutativity of t_(d), f_(D).

(2) To avoid confusion between the intended shifts and offsets, theinventor uses the term: not the offsets but “half-shifts of the exponentof the exponential function” (in the sense that the conventional shiftsare halved). One of main features in this disclosure is to design atransmitting signal so that its phase function always contains halfshifts beforehand and to cancel out the PDs based on the group-theoreticproperty of the NCP of TFSs.

c) The patent[8] discussed the preamble of the received signal as atemplate signal only in TD but was unaware of signals in FD and gave aconventional channel estimation method, being unaware of the NCP.

(1) In order to give a convergence proof of the phase updating loop(PUL) algorithm, proposed in the reference patent[6], the inventor 1)defines subspaces of the Hilbert space ε_(i), 1≤i≤4 to which thetemplate belongs; 2) shows that each of N′ TD-CCFs, CCFs of type 3 (orCCFs of type 1) yields a projection operator P₃ (or P₁) projecting ontoNMΔt- (or LΔt-)time limited (TL) Hilbert subspace ε₃ (or ε₁) and thateach of N FD-CCFs, CCFs of type 4 (or CCFs of type 2) gives a projectionoperator P₄ (or P₂) projecting onto N′M′Δf- (or LΔf-) band limited (BL)Hilbert subspace ε₄ (or ε₂); 3) proves that the PUL algorithm using thelocalization operator (LO), i.e., the von Neumann's alternativeprojection Theorem (APT) operator P₃F^(−1,d)P₄F^(d) (orP₄F^(d)P₃F^(−1,d)) for 2-D template matching on the TFP converges on theintersection of the TL Hilbert subspace and the BL Hilbert subspace(F^(−1,d),F^(d) denote IDFT and DFT); furthermore, 4) shows that theinteraction is empty by the Youla's theorem and that the PUL can giveestimates of t_(d), f_(D) with precision within time duration LΔt andbandwidth LΔf of a Gaussian chip waveform and with computationalcomplexity of but O[N+N′] of not O[N×N′], where N,N′ denote the numbersof TD- and FD-CCFs (L=(ΔtΔf)⁻¹=MM′).

(2) The MMSE algorithm for estimating the channel characteristics,proposed by the patent[8] is based on Bayes's rule and 2-parameterestimation method. Thus this algorithm has computational complexity ofO[N×N′]. While this disclosure uses a large number of likelihoodfunctional (LF)s that comprises N′ TD-LFs for estimating Doppler and NFD-LFs for estimating delay. Each LF is based on Bayes's rule. One TD-LFamong TD-LFs and FD-LF among FD-LFs in TL-TD and BL-FD Hilbertsubspaces, respectively are chosen alternatively and in the sense ofmaximum likelihood estimate (MLE). Thus the alternative algorithm withcomputational complexity of O[N+N′] is radically different fromconventional methods.

The shift estimation and received-data extraction parts 202 executes aprogram e.g., a program according to the estimation step for estimatinga time shift and a frequency shift from a received signal, as describedin (84) and its related description. The estimation step uses the TFSOof type-1 with estimated time shift and frequency shift

[MF886]

and the TFSO of type-2 with time shift and estimated frequency shift

[MF887]

and estimates a time shift and a frequency shift that are embedded inthe received signal.

The estimation step mentioned above uses 4 different types of TFSOs: aTFSO of type-1 with estimated time shift, frequency shift, and a phaseterm representing a half shift of the estimated time shift

[MF888]

a TFSO of type-2 with time shift, estimated frequency shift, and a phaseterm representing a half shift of the estimated frequency shift

[MF889]

a TFSO of type-3 with an observable time shift to be estimated, anobservable frequency shift to be estimated, and a phase termrepresenting a half shift of the time shift to be estimated

[MF890]

(or a frequency dual of the TFSO of type 3

[MF891]

),

and a TFSO of type-4 representing the estimated time shift, theestimated frequency shift, and the phase term due to the half shift ofthe estimated time shift

[MF892]

(or a frequency dual of the TFSO of type 4

[MF893]

),

and estimates a time shift and a frequency shift that are embedded inthe received signal.

The phase function of a TD-signal to be transmitted contains the phaseterm representing half shifts of one or more time shift-parameters; Thephase function of an FD-signal to be transmitted contains the phase termrepresenting half shifts of one or more frequency shift-parameters.

The receiver apparatus 200 mentioned above can realize a receivingapparatus with high efficiency. For example, when the receiver 200 isused for a radar receiver, it becomes a high rate receiving apparatus.

As described in (84) and its related description, in the estimation stepabove in the program that the shift estimation and received-dataextraction parts 202 executes, the receiver uses the CCF of type-1(C.f., MF309, (38)) that is represented by using the TFSO of type-1mentioned above

[MF894]

c _(ρ′,{right arrow over (p)}) ⁽³⁾(

_(u,i) ;{circumflex over (k)} _(d,i))

and the CCF of type-2 (C.f., MF336, (43)) that is represented by usingthe TFSO of type-2 mentioned above

[MF895]

C _(ρ,{right arrow over (p)}) ⁽⁴⁾(k _(σ,i);

_(D,i)),

refers to these CCFs, and estimates a time shift and a frequency shiftthat are embedded in the received signal as an example.

As mentioned above, the above CCF of type-1 can be represented by (c.f.,MF309, (38))

[MF896]

${{c_{\rho^{\prime}\overset{\sim}{p}}^{(3)}\left( {_{\mu};{\hat{k}}_{d}} \right)} = {{Ae}^{i\; \kappa}W^{k_{d}_{c}} \times {\sum\limits_{k \in Z}{_{k_{d},_{D}}^{d}{\psi \left\lbrack {k;\chi} \right\rbrack}\left( {W^{{\hat{k}}_{d},_{c}}_{{\hat{k}}_{d},_{p}}^{d}_{{p\; \; M},{p^{\prime}N^{\prime}M^{\prime}}}^{d}_{0,{p^{\prime}M^{\prime}}}^{d}Y_{\rho^{\prime}}^{\prime}{u_{\rho^{\prime}}^{(3)}\left\lbrack {k\text{:}Y} \right\rbrack}} \right)^{*}}}}},$

and the above CCF of type-2 can be represented by (c.f., MF336, (43))

[MF897]

${c_{\rho^{\prime}\overset{\sim}{p}}^{(4)}\left( {l_{\sigma};{\hat{l}}_{D}} \right)} = {{Ae}^{i\; \kappa}W^{k_{d}_{c}} \times {\sum\limits_{ \in Z}{_{_{D},{- k_{d}}}^{f,d}{\Psi \left\lbrack {;\chi} \right\rbrack}{\left( {W^{k_{\sigma},_{c}}_{{\hat{}}_{D},{- k_{\sigma}}}^{f,d}_{{{p\;}^{\prime}N^{\prime}M^{\prime}},{\cdots \; {pNM}}}^{f,d}_{0,{\cdots \; \rho \; M}}^{f,d}Y_{\rho}{U_{\rho}^{(4)}\left\lbrack {\text{:}Y^{\prime}} \right\rbrack}} \right)^{*}.}}}}$

In the estimation step above in the program, the receiver apparatus 1)uses the CCF of type-1 with AC shifts that is represented by using theTFSO of type-1, of type-3, and of type-4, mentioned above, (cf. MF715,(89)).

[MF898]

${{c_{p^{\prime},\overset{\_}{p},\hat{j},{\hat{j}}^{\prime}}^{{AC},{(3)}}\left( {{_{\mu};{\hat{k}}_{d}},j,j^{\prime}} \right)} = {{Ae}^{i\; \kappa}W^{k_{d}_{c}}{\sum\limits_{k \in Z}{_{k_{d},_{D}}^{d}{\psi^{AC}\left\lbrack {{k;\left\{ \chi^{(i)} \right\}_{i = 0}^{\{\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\}}},j,j^{\prime}} \right\rbrack} \times \left( {W^{{\hat{k}}_{d},_{c}}_{{\hat{k}}_{d},_{p}}^{d}W_{\mathcal{M}_{0}}^{- {\hat{j}}^{\prime}}_{k_{d}^{(j)},_{D}^{(j)}}^{d}_{{pNM},{p^{\prime}N^{\prime}M^{\prime}}}^{d}\frac{X_{\rho^{\prime}}^{{(\hat{j})}\prime}}{\sqrt{N_{1}}}{\sum\limits_{n \in I_{T}^{(\hat{j})}}{_{{nM},{\rho^{\prime}M^{\prime}}}^{d}X_{n}^{(\hat{j})}{g\lbrack k\rbrack}}}} \right)^{*}}}}},$

and the above CCF of type-2 with AC shifts that is represented by usingthe TFSO of type-2, of type-3, and of type-4, mentioned above, (cf.MF715, (90)).

[MF899]

${{c_{p,\overset{\_}{p},\hat{j},{\hat{j}}^{\prime}}^{{AC},{(4)}}\left( {{k_{\sigma};{\hat{}}_{D}},j,j^{\prime}} \right)} = {{Ae}^{i\; \kappa}W^{k_{a}_{c}}{\sum\limits_{ \in Z}{_{_{D},{\cdots }_{D}}^{f,d}{\Psi^{AC}\left\lbrack {{;\left\{ \chi^{(i)} \right\}_{i = 0}^{\{\frac{\mathcal{M} - 1}{\mathcal{M}_{0}}\}}},j,j^{\prime}} \right\rbrack} \times \left( {W^{k_{\sigma},_{c}}_{{\hat{}}_{D},{\cdots \; k_{\sigma}}}^{f,d}W_{\mathcal{M}_{0}}^{- {\hat{j}}^{\prime}}_{_{D}^{(j)},{\cdots \; k_{d}^{(\hat{j})}}}^{d}_{{p^{\prime}N^{\prime}M^{\prime}},{- {pNM}}}^{f,d}\frac{X_{\rho}^{(\hat{j})}}{\sqrt{N_{1}^{\prime}}}{\sum\limits_{n^{\prime} \in I_{F}^{(\hat{j})}}{_{{n^{\prime}M^{\prime}},{{- \rho}\; M}}^{f,d}X_{n^{\prime}}^{{(\hat{j})}\prime}{G\lbrack \rbrack}}}} \right)^{*}}}}},$

2) refers to these CCFs, and 3) estimates a time shift and a frequencyshift that are embedded in the received signal as an example.

As described above, the TD-CCF of type-1 with AC shifts (see MF715,(89)) and the FD-CCF of type-2 with AC shifts (see MF715, (90)) containthe PD due to the NCP of modulation and demodulation by the carrierf_(c) (or l_(c)), accompanied by the delay t_(d) (or k_(d))

[MF900]

e ^(i2πt) ^(d) ^(f) ^(c)

or its discretized version

[MF901]

and the PD due to the phase term e^(iκ) of the attenuation factorAe^(iκ).

As described in MF400, (cf. (60), (61)) and in its related description,the shift estimation and received-data extraction parts 202 updates apair of the estimated delay and estimated Doppler

[MF902]

({circumflex over (k)} _(d,s),

_(D,s+1))

or

[MF903]

({circumflex over (k)} _(d,s+1),

_(D,s))

by using the recursion formula for the CP (the AC shift version of (60),MF400, but the AC shift version of the formula for the OP (61) isomitted)

[MF904]

$\begin{matrix}{\left. \begin{matrix}{{\left( {\rho^{\prime,*},_{\mu}^{*},{\hat{j}}^{*},{\hat{j}}^{\prime,*}} \right) = {{argmax}_{\rho^{\prime},_{\mu},\hat{j},{\hat{j}}^{\prime}}\frac{\sqrt{{PP}^{\prime}}{c_{\rho^{\prime},\overset{\_}{p},\hat{j},{\hat{j}}^{\prime}}^{{AC},{(3)}}\left( {{_{\mu};{\hat{k}}_{d}},j,j^{\prime}} \right)}}{X_{\rho^{\prime}}^{\prime}{\hat{A}\left( \theta_{s}^{\prime,d} \right)}e^{i\; {\hat{\kappa}{(\theta_{s}^{\prime,d})}}}}}},} \\{\left( {\rho^{*},k_{\sigma}^{*},{\hat{j}}^{*},{\hat{j}}^{\prime,*}} \right) = {{argmax}_{\rho,k_{\sigma},\hat{j},{\hat{j}}^{\prime}}\frac{\sqrt{{PP}^{\prime}}{C_{\rho,\overset{\_}{p},\hat{j},{\hat{j}}^{\prime}}^{{AC},{(4)}}\left( {{k_{\sigma};{\hat{}}_{D}},j,j^{\prime}} \right)}}{X_{\rho}{\hat{A}\left( \theta_{s}^{\prime,d} \right)}e^{i\; {\hat{\kappa}{(\theta_{s}^{\prime,d})}}}}}}\end{matrix} \right\} \mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {CP}} & (109)\end{matrix}$

and sets the pair of the MLEs

[MF905]

(

*_(μ) ,k* _(σ))

to the the pair of two estimates

[MF906]

({circumflex over (k)} _(d,s+1),

_(D,s+1)).

Furthermore, executions in the shift estimation and received-dataextraction parts 202 are

1) to refer to the TD-CCF of type-1 with AC shifts (cf. MF715, MF898,(89))

[MF907]

c _(ρ′,{right arrow over (p)}) ^(AC,(3))(

_(μ) ;{circumflex over (k)} _(d) ,j,j′)

whose real part is maximized by varying

_(μ); 2) to determine the MLE

[MF908]

*_(μ)

and 3) to set it to

[MF909]

_(D,s+1).

Other executions in the parts 202 are 1) to refer to the FD-CCF oftype-2 with AC shifts (cf. MF715, MF899, (90))

[MF910]

C _(ρ,{right arrow over (ρ)}) ^(AC,(4))(k _(σ);

_(D) ,j,j′)

whose real part is maximized by varying k_(σ); 2) to determine the MLE

[MF911]

k* _(σ)

and 3) to set it to

[MF912]

{circumflex over (k)} _(d,s+1).

Thus the execution in the shift estimation and received-data extractionparts 202 is to alternatively update estimates of a time shift and afrequency shift.

The above estimation step in the shift estimation and received-dataextraction parts 202 estimates the time shift and the frequency shiftthat are embedded in the signal, with use of MLEs of frequency shift fordetecting N′ TD-templates and MLEs of time shift for detecting NFD-templates.

Furthermore, in the above estimation step in the shift estimation andreceived-data extraction parts 202, when referring to the TD-CCF oftype-1 with AC shifts (cf. MF715, MF898, (89))

[MF913]

c _(ρ′,{right arrow over (p)}) ^(AC,(3))(

_(μ) ;{circumflex over (k)} _(d) ,j,j′)

and the FD-CCF of type-2 with AC shifts (cf. MF715, MF899, (90))

[MF914]

C _(ρ,{right arrow over (p)}) ^(AC,(4))(k _(σ);

_(D) ,j,j′),

the receiver apparatus firstly replaces the rhs of the first equation of(60)

[MF915]

c _(ρ′,{right arrow over (p)}) ⁽³⁾

and the rhs of the second equation of (60)

[MF916]

C _(ρ,{right arrow over (p)}) ⁽⁴⁾

by the associated CCFs with AC shifts and secondly applies the argmaxoperation to those CCFs with AC shifts, and thirdly augments variablesof the argmax operation by a pair of two estimated integers(cf. MF904,(109))

[MF917]

(ĵ,ĵ′).

The receiving apparatus in the above estimation step in the shiftestimation and received-data extraction parts 202 is a receiver part ofreceiving a signal

[MF918]

TD-CEψ^(AC)[k]/FD-CEΨ^(AC)[

]

that is the output signal of the j-th AC with delay and Doppler shifts

[MF919]

(k _(d) ^((j)),

_(D) ^((j)))

having an integer j, one of two encoded integers by an encoder of k for

[MF920]

-ary communication.

In accordance with an embodiment of the present invention, as discussedabove, the receiving apparatus that is an apparatus of receiving asignal comprises an estimation part for estimating a time shift and afrequency shift that are embedded in the signal, with reference to anon-commutative shift parameter space of co-dimension 2.

Furthermore, in the MLE, the transmitting TD- and FD-CE signals (87) (or(27)) (according to the state that the switches 1-1,1-2 in thetransmitter of FIG. 12 are on the downward or upward side, respectively,the degree of multiplexing is varied) are given by multiplexing theirassociated TD- and FD-signatures (88) (or (25)). While the signaturesthemselves are obtained by 2-D BPSK modulating a Gaussian function andits FD function by 2-D PCs. As shown in (87), the transmitting TD- andFD-CE signals for

[MF921]

-ary communication are obtained as follows; The TD- and FD-signaturesfirst are multiplexed using 2-D BPSK modulation by the j-th PC chosenamong the set of independent 2-D PCs

[MF922]

$\left\{ {\chi^{(i)} = \left( {\left\{ X_{m}^{(i)} \right\},\left\{ X_{m^{\prime}}^{\prime {(i)}} \right\}} \right)} \right\}_{i = 0}^{\{\frac{\mathcal{M} - 1}{\mathcal{M}}\}}$

according to the encoded integer pair of k

[MF923]

${j = \left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack},{j^{\prime} = \left\{ \frac{k}{\mathcal{M}_{0}} \right\}},$

and secondly, are passed to the AC channel with its associated time andfrequency shifts

[MF924]

(k _(d) ^((j)),

_(D) ^((j)))

or equivalently the transmitter makes the associated TFSOs

[MF925]

(or

)

act on the multiplexed TD- or FD-signature.

In accordance with an embodiment of the present invention, as discussedabove, the transmitting method that is a method for transmitting asignal to be transmitted comprises a shift step for time-frequencyshifting the signal to be transmitted, with reference to anon-commutative shift parameter space of co-dimension 2.

Furthermore, in the above MLE step, when using the CCFs in (89), (90)that are cross-correlations between the received signal

[MF926]

TD-CEψ^(AC)[k]/FD-CEΨ^(AC)[

],

being passed through the AC channel with its associated time andfrequency shifts

[MF927]

(k _(d) ^((j)),

_(D) ^((j))),

characterized by the encoder of k for

[MF928]

-ary communication and an estimated received template (the first andsecond terms of the rhss of (89), (90) indicate a transmitting signaland an estimated received template, respectively), one should notice thefact that the symbol Σ in (87)-(90) implies the multiplexing, beingequivalent to the non-overlapped superposition of a signal, and thatsuch multiplexing can be simulated by using several kinds of TFSOs. Inthe generation process of a transmitting signal in FIG. 12, severalkinds of multiplexing are performed. In (87), (88), TD- andFD-signatures are obtained by multiplexing 2-D BPSK modulated TD- andFD-chip waveforms by 2-D PCs

[MF929]

χ^((i))=({X _(m) ^((i)) },{X′ _(m′) ^((i))})

respectively of periods

[MF930]

${N_{1} = {N\text{/}\sqrt{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1}}},{N_{1}^{\prime} = {N^{\prime}\text{/}{\sqrt{\left\lbrack \frac{\mathcal{M} - 1}{\mathcal{M}_{0}} \right\rbrack + 1}.}}}$

The transmitting TD- and FD-CE are obtained by time-frequency shiftingthe resultant TD- and FD-signatures by data level-shifts andmultiplexing those signatures. If the superposition of signals is simplydone without the use of time and frequency shifts, then the detection oftemplates is difficult; One should pay the price for the PDs due toshift operations: It is, however, obvious that fortunately, the PDs arevaluable clues in detecting templates and thus the TFSOs are essentialtools for precise evaluations of PDs and canceling out the PDs.Furthermore, the rhss of (89), (90) contain the PD due to modulation anddemodulation by the carrier

_(c), accompanied by the delay k_(d)

[MF931]

and the compensation

[MF932]

₀-ary

for canceling out the PD due to the phase modulation by the encodedinteger j′ of k

[MF933]

${\mathcal{M}_{0}\text{-}{ary}\mspace{14mu} W_{\mathcal{M}_{0}}^{- j^{\prime}}},{j^{\prime} = {\left\{ \frac{k}{\mathcal{M}_{0}} \right\}.}}$

As shown above, the shift step generates the signal to be transmitted byPSK-modulating a chip pulse wave in the TD by the TD-PC of period N andPSK-modulating the TD-PSK modulated TD-pulse wave by the FD-PC of periodN′ to generate a multicarrier version of the TD-PSK modulated TD-pulsewave that is a transmitting signal.

In accordance with an embodiment of the present invention, in thetransmitting apparatus in accordance with an embodiment of the presentinvention, the transmitter, when the switches 1-1,1-2 are on downward inFIG. 12, first PSK modulates a TD-chip waveform by a TD-PC of period Nand secondly PSK modulates the PSK modulated signal by an FD-PC ofperiod N′, thirdly multiplexes the doubly-modulated multi-carrier signalby data symbols, and fourthly passes the data-multiplexed signal throughthe AC with shifts

[MF934]

(k _(d) ^((j)),

_(D) ^((j))),

having the encoded integer j of k

[MF935]

$j = {\left\lbrack \frac{k}{\mathcal{M}_{0}} \right\rbrack.}$

The output signal of the AC gives

[MF936]

TD-CEψ^(AC)[k]/FD-CEΨ^(AC)[

].

In accordance with an embodiment of the present invention, in thereceiver apparatus in accordance with an embodiment of the presentinvention, the receiver, when the switches 2-1,2-2 switch on downward inFIG. 13, the receiver performs maximization of the real parts of theCCFs (cf. MF715, (89), (90)) in terms of 4 variables of the argmaxoperation

[MF937]

(ρ′,{right arrow over (p)},ĵ,ĵ′)(or (ρ′,{right arrow over (p)},ĵ,ĵ′)).

If the PUL algorithm for updating MLEs

[MF938]

(k* _(σ),

*_(μ))

converges, then the data symbol

[MF939]

d _({right arrow over (q)})

can be recovered and the decoding of k can be done.

In accordance with an embodiment of the present invention, thegeneralization of several apparatuses with applications to images insubsection 8.3 is a natural 2-D extension of 1-D signal, exploiting thehalf shifts due to the NCP of the space-spatial frequency shift (SSFS)s(see (102)-(108)).

(An Extended Version of an Embodiment for Images)

In accordance with an embodiment of the present invention, thisdisclosure gives a method for receiving an image that comprises anestimation step for estimating a space shift and a spatial frequencyshift that are embedded in the received image signal, with reference toa parameter space of co-dimension 2, wherein each of the space shift andthe spatial frequency shift has dimension ≥2.

(An Extended Version of an Embodiment for Images)

In accordance with an embodiment of the present invention, the receiverin the above estimation step refers to the 2×2-D symmetrical SSFSoperator (SSFSO)s

[MF940]

_(a) _(i) _(,b) _(i) ^((2),d)

and its frequency dual

[MF941]

_(b) _(i) _(,−a) _(i) ^((2),f,d)

and estimates space shifts and spatial frequency shifts.

(An Extended Version of an Embodiment for Images)

In accordance with an embodiment of the present invention, thisdisclosure gives a method for transmitting an image signal, comprising ashift step for space-spatial frequency shifting the image signal to betransmitted, with reference to a parameter space, wherein each of thespace shift and the spatial frequency shift has dimension ≥2.

(An Extended Version of an Embodiment for Images)

In accordance with an embodiment of the present invention, thetransmitter in the above estimation step refers to the 2×2-D symmetricalSSFS operator (SSFSO)s

[MF942]

_(a) _(i) _(,) _(i) ^((2),d)

and its frequency dual

[MF943]

_(b) _(i) _(,−a) _(i) ^((2),f,d)

and space- and spatial frequency-shifts an image.

(The Received-Data Output Part 203)

The received-data output part 203 outputs the extracted data by theshift estimation and received-data extraction parts 202

<The Signal Flow-Chart Between a Transmitter and a Receiver>

FIG. 19 shows a flow-chart of transmitter-receiver using thetransmitter-receiver system with the number 1.

(S101)

In the step S101, the data acquisition part 101 gets data to betransmuted. Concrete procedures by the part 101 have been describedabove.

(S102)

In the step S102, the signal generation part 102 generates a signal tobe transmitted. Concrete procedures by the part 102 have been describedabove.

(S103)

In the step S103, the transmitter part 103 transmits the signal to betransmuted. Concrete procedures by the part 103 have been describedabove.

(S201)

In the step, the receive part 201 receives a transmitted signal by thetransmitter part S103. Concrete procedures by the part 201 have beendescribed above.

(S202)

In the step S202, the shift estimation and received-data extractionparts 202 estimates shifts and extracts received data, simultaneously.Concrete procedures in the shift estimation and received-data outputpart 202 are as described above.

(S203)

In the step S203, the received-data output parts 203 outputs thereceived data extracted by the shift estimation and received-dataextraction parts 202. Concrete procedures in the received-data outputpart 203 are as described above.

<<Example 2 of Embodiments of Communication Systems>>

The example 1 of embodiments of communication systems that is based onthe theoretical basis as explained above is explained, together withcited figures.

FIG. 20 shows a block-diagram representing the transmitter-receiversystem with the number 1 a attached to its surrounding dotted line,according to an embodiment of the present invention. As shown in FIG.20, the system with the number 1 a comprises a transmitter apparatus 100a and a receiver apparatus 200 a. The transmitter-receiver system withthe number 1 a, like the transmitter-receiver system with the number 1in FIG. 18 as discussed above, can be used for both a radar and acommunication system (e.g., transmitter-receiver system to convey voicedata and or image data). Other parts are labelled with the same numberas their associated parts in the transmitter-receiver system with thenumber 1 and hence no explanation for them is necessary.

<Transmitter Apparatus 100 a >

The transmitter apparatus 100 a is an apparatus that executes a programimposed on a transmitter as described in the above description.

As shown in FIG. 20, the transmitter apparatus 100 a comprises theacquisition part of transmitting data 101, the generator part of atransmitting signal 102 a, and the transmitter part 103 as an example.

(The Transmitting Signal Generation Part 102 a)

the transmitting signal generation part 102 a comprises the generatorpart of a transmitting signal 102 relating to the example 1 ofembodiments of transmitter-receiver system and the embedding shift part110 as well.

As discussed in (84) just below, the embedding shift part 110 generatesa transmitting signal into which a set of N_(path) delay and Dopplerparameters

[MF944]

{(k _(d,i)

_(D,i))}_(i=1) ^(N) ^(path) .

are embedded.

The generator part of a transmitting signal 102 a time-frequency shiftsa signal to be transmitted by using one time shift (or multiple timeshifts) and one frequency shift (or multiple frequency shifts)

[MF945]

{(k _(d,i)

_(D,i))}_(i=1) ^(N) ^(path) .

<Receiver Apparatus 200 a>

The receiver apparatus 200 a is an apparatus that executes a programimposed on a receiver as described in the above description.

As shown in FIG. 20, the receiver apparatus 200 a comprises a receiverpart 201, a shift estimation and received-data extraction parts 202 a,and a received-data decision part 203 as an example

(The Shift Estimation and Received-Data Extraction Parts 202 a)

The shift estimation and received-data extraction parts 202 a comprisesthe parts similar as parts that the shift estimation and received-dataextraction parts 202 comprises, relating to the example 1 of embodimentsof transmitter-receiver system as an example.

<The Signal Flow-Hart Between a Transmitter and a Receiver>

FIG. 21 shows a flow-chart of transmitter-receiver using thetransmitter-receiver system with the number 1 a. The steps S101,S103,S201, and S203 are same processing as the description using FIG. 19and hence no explanation for them is necessary.

(S102 a)

In the step S102 a, the generator part of a transmitting signal 102 awith the embedding-shift part 110 generates a transmitting signal.Concrete procedures by the generator part of a transmitting signal S102a have been described above.

(S202 a)

In the S202 a, the shift estimation and received-data extraction parts202 a estimates shifts and extracts the received data, simultaneously.Concrete procedures in the shift estimation and received-data extractionparts 202 a are as described above.

<<Example 3 of Embodiments of Communication Systems>>

The transmitter-receiver system that is explained by referring to FIGS.18-21 may be an embodiment that executes programs as described insubsection <8.3 signal processing exploiting multi-dimensionalnon-commutative apatial-spatial frequency shifts>:

As described in subsection <8.3 signal processing exploitingmulti-dimensional non-commutative spatial-spatial frequency shifts>, thereceived apparatus 200 executes the estimation step for estimating SSFSsthat are embedded in the received images by referring to the parameterspace, i.e., SSFS parameter space of co-dimension ≥2 as an example.

As described in subsection <8.3 signal processing exploitingmulti-dimensional non-commutative spatial-spatial frequency shifts>, theabove estimation step, the receiver refers to the 2×2-D symmetricalspatial shift and spatial frequency shift (SSFS) operator (SSFSO)srepresenting the half-shift-PDs due to the NCP of the SSFSs

[MF946]

_(a) _(i) _(,b) _(i) ^((2),d)

and its frequency dual

[MF947]

_(b) _(i) _(,−a) _(i) ^((2),f,d)

and estimates SSFSs.

Similarly, in accordance with an embodiment of the present invention,the transmitter, i.e., the method for transmitting an image refers tothe SSFS parameter space of co-dimension ≥2 and comprises 2 differentSSFSOs that represent spatial-spatial frequency shifting an image to betransmitted and the PD due to the half-shift of the spatial frequencyshift (or the PD due to the half-shift of the spatial shift).

In the above shift step, the transmitter refers to the symmetricalSSFSOs

[MF948]

_(a) _(i) _(,b) _(i) ^((2),d)

and its frequency dual

[MF949]

_(b) _(i) _(,−a) _(i) ^((2),f,d)

and estimates SSFSs.

[Example of Embodiments Using Software]

The control-block in the transmitter apparatuses 100, 100 a and thereceiver apparatuses 200,200 a (in particular, the generator parts of atransmitting signal 102, 102 a and the shift estimation andreceived-data extraction parts 202,202 a) may be realized by logicalcircuits implemented by integrated circuit (IC chip)s or by software.

In the case of the implementation by software, the transmitterapparatuses 100, 100 a and the receiver apparatuses 200,200 a areequipped with a computer that executes programs for each of severalfunctions; e.g., this computer comprises multiple processors andcomputer-readable memories. The computer executes the programs that theprocessors read from the memories so that it can perform the aim of thepresent invention. non-transitory memory, e.g, read-only-memory (ROM),tape, disc, card, IC memory, and programmable logical circuits, etc areused for the memories. The computer may be equipped with random accessmemory (RAM) performing the above programs. The above programs may beinstalled via any transmission media (such as communication network,broadcast, radio wave) and supplied by the above computer. In accordancewith an embodiment of the present invention, the above program that maybe embodied in an embedded data signal into a carrier wave by electronictransmission and may be realized.

Program products by computer that realize several functions of thetransmitter apparatuses 100, 100 a and the receiver apparatuses 200,200a include one of embodiments of the present invention. The above programproduct by computer loads programs that are provided by any transmissionmedia by using at least one computer and makes the computer to executeat least one program. Thus, at least one processor associated with thecomputer executes at least one program. Thus each of functions of thetransmitter apparatuses 100, 100 a and the receiver apparatuses 200,200can be realized. The program product by computer makes at least one ofprogram-installed computers to execute steps in transmittingprocess(method of transmission) and receiving process(method ofreceiving).

In accordance with an embodiment of the present invention, the abovementioned embodiments have no limitation. The embodiments are modifiedwithin the claims. Comprising several methods or techniques given indifferent embodiments leads to an embodiment of the present patent andincludes one of methods of the present patent. Furthermore, adequatecomprising several methods or techniques given in different embodimentsmay provide a new technical characteristic feature.

INDUSTRIAL APPLICATION

The present patent is ideally suited to wireless transmitter-receivercommunication systems and to radar systems.

LIST OF SYMBOLS FOR PARTS OF EMBODIMENT

-   1,1 a: transmitter-receiver system-   100,100 a: transmitter apparatus-   101: acquisition part of transmitting data-   102,102 a: transmitting signal generation part-   103: transmitter part-   110: embedding-shift part-   200,200 a: receiver apparatus-   201: receiver part-   202,202 a: shift estimation and received-data extraction parts    (estimation part)-   203: received-data decision part

1. A method for receiving a signal, comprising an estimation step forestimating a time shift and a frequency shift that are embedded in thesignal, wherein the estimation step refers to a non-commutative shiftparameter space of co-dimension
 2. 2. The method recited in claim 1,wherein the non-commutative shift parameter space of co-dimension 2 is aspace obtained by 3-dimensionalizing a plane, which is defined by afirst axis indicative of time and a second axis indicative of frequency,by attaching, to the plane, a third axis indicative of time-frequencyshifts.
 3. The method recited in claim 1, wherein the estimation stepestimates the time shift and the frequency shifts that are embedded inthe signal, with use of a time-frequency shift operator (TFSO) of type 1representing an estimated time shift, frequency shift, and a phase termdue to a half shift of the estimated time shift [MFC1]

, a TFSO of type 2 representing time shift, an estimated frequencyshift, and a phase term due to a half shift of the estimated frequencyshift [MFC2]

a TFSO of type 3 representing an observable time shift to be estimated,an observable frequency shift to be estimated, and a phase term due to ahalf shift of the time shift to be estimated [MFC3]

or a frequency dual of the TFSO of type 3 [MFC4]

and a TFSO of type 4 representing the estimated time shift, theestimated frequency shift, and the phase term due to the half shift ofthe estimated time shift [MFC5]

or a frequency dual of the TFSO of type 4 [MFC6]


4. The method recited in claim 3, wherein the estimation step estimatesthe time shift and the frequency shift that are embedded in the signal,with use of a time-domain-cross-correlation function (TD-CCF) of type 1,represented in terms of the TFSOs of type 1, of type 3, and of type 4,and a frequency-domain-cross-correlation function (FD-CCF) of type 2,represented in terms of the TFSOs of type 2, of type 3, and of type 4.5. The method recited claim 4, wherein the TD-CCF of type 1 and theFD-CCF of type 2 contain a phase distortion (PD) due to anon-commutative shift operation between modulation and demodulation by acarrier f_(c) and a time shift t_(d) [MFC7]e ^(i2πt) ^(d) ^(f) ^(c) or a discretized version of the PD [MFC8]

, and a PD due to the phase term e^(iκ) of the attenuation factorAe^(iκ).
 6. The method recited in claim 4, wherein the estimation stepincludes two alternative updating steps: (1) updating the estimatedfrequency shift with reference to the TD-CCF of type 1 and (2) updatingthe estimated time shift with reference to the FD-CCF of type
 2. 7. Themethod recited in claim 1, wherein the estimation step estimates thetime shift and the frequency shift that are embedded in the signal, withuse of maximum-likelihood estimate (MLE)s of frequency shift fordetecting N′ time-domain (TD)-templates and MLEs of time shift fordetecting N frequency-domain (FD)-templates.
 8. A receiving apparatusfor receiving a signal, comprising an estimation part for estimating atime shift and a frequency shift that are embedded in the signal, withreference to a non-commutative shift parameter space of co-dimension 2.9. A method for transmitting a signal, comprising a shift step fortime-frequency shifting the signal to be transmitted, with reference toa non-commutative shift parameter space of co-dimension
 2. 10. Themethod recited in claim 9, wherein the shift step generates the signalto be transmitted by phase-shift-keying (PSK)-modulating a chip pulsewave in the time domain (TD) by the TD-phase code (PC) of period N andPSK-modulating the TD-PSK modulated TD-pulse wave by the FD-PC of periodN′ to generate a multicarrier version of the TD-PSK modulated TD-pulsewave.
 11. A transmitter apparatus for transmitting a signal, comprisinga time-frequency shift part for time-frequency shifting the signal to betransmitted, with reference to a non-commutative shift parameter spaceof co-dimension
 2. 12. A transmitter-receiver system, comprising: atransmitter apparatus; and a receiver apparatus, the transmitterapparatus including a time-frequency shift part for time-frequencyshifting a signal to be transmitted, with reference to a non-commutativeshift parameter space of co-dimension 2, the receiver apparatusincluding an estimation part for estimating a time shift and a frequencyshift that are embedded in the signal to be transmitted, with referenceto the non-commutative shift parameter space of co-dimension
 2. 13. Amethod for receiving an image signal, comprising an estimation step forestimating a space shift and a spatial frequency shift (SSFS) that areembedded in the image signal, with reference to a parameter space ofco-dimension 2, wherein each of the space shift and the spatialfrequency shift has dimension ≥2.
 14. The method recited in claim 13,wherein the estimation step estimates the SSFSs, with reference to 2-Dsymmetrical SSF operator (SSFSO)s [MFC9]

_(a) _(i) _(,b) _(i) ^((2),d) or a frequency dual of the 2-D symmetricalSSFSOs [MFC10]

_(b) _(i) _(,−a) _(i) ^((2),f,d).
 15. A method for transmitting an imagesignal, comprising a shift step for space-spatial frequency shifting theimage signal to be transmitted, with reference to a parameter space,wherein each of the space shift and the spatial frequency shift hasdimension ≥2.
 16. The method recited in claim 15 for receiving a signal,wherein the shift step space-spatial frequency shifts the image signalto be transmitted, with reference to 2-D symmetrical SSFSOs [MFC9]

_(a) _(i) _(,b) _(i) ^((2),d) and a frequency dual of the 2-Dsymmetrical SSFSOs [MFC10]

_(b) _(i) _(,−a) _(i) ^((2),f,d).